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1.2 algebraic expressions

The document discusses expressions and polynomials. It provides examples of algebraic expressions and operations that can be performed on polynomials, such as factoring. Factoring polynomials is useful for easier evaluation, simplifying rational expressions, and solving equations. One example factors the polynomial 64x3 + 125 into (4x + 5)(16x2 - 20x + 25). Factoring the polynomial 2x3 - 5x2 + 2x is recommended before evaluating it for specific values of x.

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Expressions http://www.lahc.edu/math/precalculus/math_260a.html
We order pizzas from Pizza Grande. Expressions
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Expressions
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. Expressions
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Expressions
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810.
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions Definition: Mathematical expressions are calculation procedures which are written with numbers, variables, operation symbols +, –, *, / and ( )’s. If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
We order pizzas from Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions Definition: Mathematical expressions are calculation procedures which are written with numbers, variables, operation symbols +, –, *, / and ( )’s. Expressions calculate the expected future results. If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions
An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Trigonometric or log-formulas are not algebraic.
An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4,
An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 ,
An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 , (x1/2 + y)1/3 (4y2 – (x + 4)1/2)1/4
An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 , (x1/2 + y)1/3 (4y2 – (x + 4)1/2)1/4 Examples of non–algebraic expressions are sin(x), 2x, log(x + 1).
An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 , (x1/2 + y)1/3 (4y2 – (x + 4)1/2)1/4 Examples of non–algebraic expressions are sin(x), 2x, log(x + 1). The algebraic expressions anxn + an–1xn–1...+ a1x + a0 where ai are numbers, are called polynomials (in x).
An algebraic expression is a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 , (x1/2 + y)1/3 (4y2 – (x + 4)1/2)1/4 Examples of non–algebraic expressions are sin(x), 2x, log(x + 1). The algebraic expressions anxn + an–1xn–1...+ a1x + a0 where ai are numbers, are called polynomials (in x). The algebraic expressions where P and Q are polynomials, are called rational expressions. P Q
Polynomial Expressions Following are examples of operations with polynomials and rational expressions.
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – (3x – 4)(x + 5)
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – (3x – 4)(x + 5) The point of this problem is how to subtract a “product”.
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert [ ] The point of this problem is how to subtract a “product”.
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] Insert [ ] remove [ ]
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 Insert [ ] remove [ ]
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] remove [ ]
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] remove [ ]
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] remove [ ] (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = … Or distribute the minus sign and change it to an addition problem:
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] To factor an expression means to write it as a product in a non-obvious way. remove [ ] (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = … Or distribute the minus sign and change it to an addition problem:
Polynomial Expressions Following are examples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] A3 B3 = (A B)(A2 AB + B2) Important Factoring Formulas: To factor an expression means to write it as a product in a non-obvious way. A2 – B2 = (A + B)(A – B) +– +– +– remove [ ] (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = … Or distribute the minus sign and change it to an addition problem:
Example B. Factor 64x3 + 125 Polynomial Expressions A3 B3 = (A B)(A2 AB + B2)+– + –+–
Example B. Factor 64x3 + 125 64x3 + 125 = (4x)3 + (5)3 Polynomial Expressions A3 B3 = (A B)(A2 AB + B2)+– + –+– A3 B3
Example B. Factor 64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) Polynomial Expressions A3 B3 = (A B)(A2 AB + B2)+– + –+– (A B) (A2 AB + B2)+ –
Example B. Factor 64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions A3 B3 = (A B)(A2 AB + B2)+– + –+–
Example B. Factor 64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions We factor polynomials for the following purposes. A3 B3 = (A B)(A2 AB + B2)+– + –+–
Example B. Factor 64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions We factor polynomials for the following purposes. I. It’s easier to calculate an output or to check the sign of an output using the factored form. A3 B3 = (A B)(A2 AB + B2)+– + –+–
Example B. Factor 64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions We factor polynomials for the following purposes. I. It’s easier to calculate an output or to check the sign of an output using the factored form. II. To simplify or perform algebraic operations with rational expressions. A3 B3 = (A B)(A2 AB + B2)+– + –+–
Example B. Factor 64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions We factor polynomials for the following purposes. I. It’s easier to calculate an output or to check the sign of an output using the factored form. II. To simplify or perform algebraic operations with rational expressions. III. To solve equations (See next section). A3 B3 = (A B)(A2 AB + B2)+– + –+–
Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first.
Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40 Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40 Plug in x = –1: –1 [2(–1) – 1] [(–1) – 2] Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40 Plug in x = –1: –1 [2(–1) – 1] [(–1) – 2] = –1 [–3] [–3] = –9 Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40 Plug in x = –1: –1 [2(–1) – 1] [(–1) – 2] = –1 [–3] [–3] = –9 Plug in x = 3: 3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15 Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = –3/2. Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + . Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + . Determine the Signs of the Outputs. Rational Expressions We say a rational expression is in the factored form if it's numerator and denominator are factored. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + . Determine the Signs of the Outputs. Rational Expressions We say a rational expression is in the factored form if it's numerator and denominator are factored. Example E. Factor x2 – 1 x2 – 3x+ 2 It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
Example D. Determine whether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + . Determine the Signs of the Outputs. Rational Expressions We say a rational expression is in the factored form if it's numerator and denominator are factored. Example E. Factor x2 – 1 x2 – 3x+ 2 x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1) (x – 1)(x – 2) is the factored form. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
Rational Expressions We put rational expressions in the factored form in order to reduce, multiply or divide them.
Rational Expressions We put rational expressions in the factored form in order to reduce, multiply or divide them. Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e. x*y x*z = x*y x*z = y z 1
Rational Expressions We put rational expressions in the factored form in order to reduce, multiply or divide them. x*y x*z = x*y x*z = y z A rational expression that can't be cancelled any further is said to be reduced. Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e.
Rational Expressions We put rational expressions in the factored form in order to reduce, multiply or divide them. Example F. Reduce x2 – 1 x2 – 3x+ 2 x*y x*z = x*y x*z = y z A rational expression that can't be cancelled any further is said to be reduced. Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e.
Rational Expressions We put rational expressions in the factored form in order to reduce, multiply or divide them. Example F. Reduce x2 – 1 x2 – 3x+ 2 x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1) (x – 1)(x – 2) x*y x*z = x*y x*z = y z A rational expression that can't be cancelled any further is said to be reduced. factor Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e.
Rational Expressions We put rational expressions in the factored form in order to reduce, multiply or divide them. Example F. Reduce x2 – 1 x2 – 3x+ 2 x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1) (x – 1)(x – 2) x*y x*z = x*y x*z = y z A rational expression that can't be cancelled any further is said to be reduced. = (x + 1) (x – 2) factor Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e.
Rational Expressions Multiplication Rule:
Rational Expressions Multiplication Rule: P Q R S * = P*R Q*S
Rational Expressions Multiplication Rule: P Q R S * = P*R Q*S Division Rule: P Q R S ÷
Rational Expressions Multiplication Rule: P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate
Rational Expressions Multiplication Rule: P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2)
Rational Expressions Multiplication Rule: P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* Reciprocate
Rational Expressions Multiplication Rule: P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* Reciprocate To carry out these operations, put the expressions in factored form and cancel as much as possible.
Rational Expressions Multiplication Rule: To carry out these operations, put the expressions in factored form and cancel as much as possible. P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* = 2(x – 3) (y + 3) (y + 3)(y – 1) (3 – x)(3 + x)* factor and cancel
Rational Expressions Multiplication Rule: To carry out these operations, put the expressions in factored form and cancel as much as possible. P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* = 2(x – 3) (y + 3) (y + 3)(y – 1) (3 – x)(3 + x)* 1 factor and cancel
Rational Expressions Multiplication Rule: To carry out these operations, put the expressions in factored form and cancel as much as possible. P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* = 2(x – 3) (y + 3) (y + 3)(y – 1) (3 – x)(3 + x)* –1 1 factor and cancel
Rational Expressions Multiplication Rule: To carry out these operations, put the expressions in factored form and cancel as much as possible. P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* = 2(x – 3) (y + 3) (y + 3)(y – 1) (3 – x)(3 + x)* –1 1 = –2(y – 1) (x + 3)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions II. to simplify complex fractions
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 Combining Rational Expressions (LCD Method): II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 7 12 5 8 + – 16 9 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, 48 II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 7 12 5 8 + – 16 9 48 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 7 12 5 8 + – 16 94 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 67 12 5 8 + – 16 94 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 67 12 5 8 + – 16 94 3 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 67 12 5 8 + – 16 94 3 = Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 28 + 30 – 27 48 II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions The least common denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 67 12 5 8 + – 16 94 3 = Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 28 + 30 – 27 48 = 48 31 II. to simplify complex fractions III. to solve rational equations (later)
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3Example I. Combine
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Example I. Combine
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), Example I. Combine
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, Example I. Combine
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y2 + 2y – 3)(y – 1)(y + 2) 2y – 1 y – 3[ ] Example I. Combine
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y2 + 2y – 3)(y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) Example I. Combine LCD
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) Example I. Combine LCD expand and simplify. (y – 1)(y + 3)
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) (y + 3) Example I. Combine LCD expand and simplify. (y – 1)(y + 3)
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) (y + 3) (y + 2) Example I. Combine LCD expand and simplify. (y – 1)(y + 3)
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) = (2y – 1)(y + 3) – (y – 3)(y + 2) (y + 3) (y + 2) Example I. Combine LCD LCD expand and simplify. (y – 1)(y + 3)
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) = (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3 (y + 3) (y + 2) Example I. Combine LCD LCDLCD expand and simplify. (y – 1)(y + 3)
Rational Expressions – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) = (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3 So – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 = y2 + 6y + 3 (y – 1)(y + 2)(y + 3) (y + 3) (y + 2) Example I. Combine LCD LCDLCD expand and simplify. (y – 1)(y + 3)
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. 2 3 –4 1 3 2
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 –4 1 3 2
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Their LCD is 12.
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 Their LCD is 12.
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 Their LCD is 12. = 1
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4 6
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4 6 3 4
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4 6 3 4 = 3 4 – 18 – 8
Rational Expressions Example J. Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4 6 3 4 = 3 4 – 18 – 8 = 14 5
Rational Expressions Example K. Simplify – (x – h) 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. (x + h) 1 2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. – (x – h) 1 (x + h) 1 2h = – (x – h) 1 (x + h) 1 2h (x + h)(x – h)[ ] (x + h)(x – h)* = –(x + h) (x – h) 2h(x + h)(x – h) = 2h 2h(x + h)(x – h) = 1 (x + h)(x – h)
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. Rationalize Radicals
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals
Example K: Rationalize the numerator To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. h x + h – x Rationalize Radicals
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * Example K: Rationalize the numerator h x + h – x
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * = h (x + h)2 – (x)2 (x + h + x) Example K: Rationalize the numerator h x + h – x
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * = h (x + h)2 – (x)2 (x + h + x) Example K: Rationalize the numerator h x + h – x (x + h) – (x) = h
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * = h (x + h)2 – (x)2 (x + h + x) = h h (x + h + x) Example K: Rationalize the numerator h x + h – x (x + h) – (x) = h
To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * = h (x + h)2 – (x)2 (x + h + x) = h h (x + h + x) = 1 x + h + x Example K: Rationalize the numerator h x + h – x (x + h) – (x) = h
Exercise A. Factor each expression then use the factored form to evaluate the given input values. No calculator. Applications of Factoring 1. x2 – 3x – 4, x = –2, 3, 5 2. x2 – 2x – 15, x = –1, 4, 7 3. x2 – x – 2, x = ½ ,–2, –½ 4. x3 – 2x2, x = –2, 2, 4 5. x4 – 3x2, x = –1, 1, 5 6. x3 – 4x2 – 5x, x = –4, 2, 6 B. Determine if the output is positive or negative using the factored form. 7. x2 – 4 x + 4 8. x3 – 2x2 x2 – 2x + 1 , x = –3, 1, 5 , x = –0.1, 1/2, 5 4. x2 – 4 x + 4 5. x2 + 2x – 3 x2 + x 6. x3 – 2x2 x2 – 2x + 1 , x = –3.1, 1.9 , x = –0.1, 0.9, 1.05 , x = –0.1, 0.99, 1.01 1. x2 – 3x – 4, x = –2½, –2/3, 2½, 5¼ 2. –x2 + 2x + 8, x = –2½, –2/3, 2½, 5¼ 3. x3 – 2x2 – 8x, x = –4½, –3/4, ¼, 6¼,
C. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions 1. 10x * 2 5x3 15x 4 * 16 25x42. 3.12x6 * 5 6x14 4. 75x 49 * 42 25x3 5. 2x – 4 2x + 4 5x + 10 3x – 6 6. x + 4 –x – 4 4 – x x – 4 7. 3x – 9 15x – 5 3 – x 5 – 15x 8. 42 – 6x –2x + 14 4 – 2x –7x + 14 * * * * 9. (x2 – x – 2 ) (x2 – 1) (x2 + 2x + 1) (x2 + x ) * 10. (x2 + 5x – 6 ) (x2 + 5x + 6) (x2 – 5x – 6 ) (x2 – 5x + 6) * 11. (x2 – 3x – 4 ) (x2 – 1) (x2 – 2x – 8) (x2 – 3x + 2) * 12. (– x2 + 6 – x ) (x2 + 5x + 6) (x2 – x – 12) (6 – x2 – x) * 13.(3x2 – x – 2) (x2 – x + 2) (3x2 + 4x + 1) (–x – 3x2) 14. (x + 1 – 6x2) (–x2 – 4) (2x2 + x – 1 ) (x2 – 5x – 6) 15. (x3 – 4x) (–x2 + 4x – 4) (x2 + 2) (–x + 2) 16. (–x3 + 9x ) (x2 + 6x + 9) (x2 + 3x) (–3x2 – 9x) ÷ ÷ ÷ ÷
Multiplication and Division of Rational Expressions D. Multiply, expand and simplify the results. 1. x + 3 x + 1 (x2 – 1) 2. x – 3 x – 2 (x2 – 4) 3. 2x + 3 1 – x (x2 – 1) 4. 3 – 2x x + 2 (x + 2)(x +1) 5. 3 – 2x 2x – 1 (3x + 2)(1 – 2x) 6. x – 2 x – 3 ( x + 1 x + 3)( x – 3)(x + 3) 7. 2x – 1 x + 2 ( – x + 2 2x – 3 )( 2x – 3)(x + 2) + 8. x – 2 x – 3 ( x + 1 x + 3 ) ( x – 3)(x + 3)– 9. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) 10. x + 3 x2 – 4 ( – 2x + 1 x2 + x – 2 ) ( x – 2)(x + 2)(x – 1) 11. x – 1 x2 – x – 6 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 2)(x + 1)
E. Combine and simplify the answers. –3 x – 3 + 2x –6 – 2x 3. 2x – 3 x – 3 – 5x + 4 5 – 15x 4. 3x + 1 6x – 4 – 2x + 3 2 – 3x5. –5x + 7 3x – 12+ 4x – 3 –2x + 86. 3x + 1 + x + 3 4 – x211. x2 – 4x + 4 x – 4 + x + 5 –x2 + x + 2 12. x2 – x – 6 3x + 1 + 2x + 3 9 – x213. x2 – x – 6 3x – 4 – 2x + 5 x2 – x – 6 14. –x2 + 5x + 6 3x + 4 + 2x – 3 –x2 – 2x + 3 15. x2 – x 5x – 4 – 3x – 5 1 – x216. x2 + 2x – 3 –3 2x – 1 + 2x 2 – 4x 1. 2x – 3 x – 2 + 3x + 4 5 – 10x 2. 3x + 1 2x – 5 – 2x + 3 5 – 10x 9. –3x + 2 3x – 12 + 7x – 2 –2x + 8 10. 3x + 5 3x –2 – x + 3 2 – 3x7. –5x + 7 3x – 4 + 4x – 3 –6x + 88. Addition and Subtraction of Fractions
Complex Fractions 1 2x + 1 – 2 3 – 1 2x + 1 3. –2 2x + 1 – + 3 x + 4 4. 1 x + 4 2 2x + 1 4 2x + 3 – + 3 x + 4 5. 3 3x – 2 5 3x – 2 –5 2x + 5 – + 3 –x + 4 6. 2 2x – 3 6 2x – 3 2 3 + 2 2 – – 1 6 2 3 1 2+ 1. 1 2 – + 5 6 2 3 1 4 – 2. 3 4 3 2 + F. Combine and simplify the answers. 7. 2 x – 1 – + 3 x + 3 x x + 3 x x – 1 8. 3 x + 2 – + 3 x + 2 x x – 2 x x – 2 9. 2 x + h – 2 x h 10. 3 x – h – 3 x h 11. 2 x + h – 2 x – h h 12. 3 x + h – h 3 x – h
G. Rationalize the denominator. 1. 1 – 3 1 + 3 2. 5 + 2 3 – 2 3. 1 – 33 2 + 3 4. 1 – 53 4 + 23 5. 32 – 33 22 – 43 6. 25 + 22 34 – 32 7. 42 – 37 22 – 27 8. x + 3 x – 3 9. 3x – 3 3x + 2 10. x – 2 x + 2 + 2 11. x – 4 x – 3 – 1 Algebra of Radicals

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1.2 algebraic expressions

  • 1.
  • 2.
    We order pizzasfrom Pizza Grande. Expressions
  • 3.
    We order pizzasfrom Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Expressions
  • 4.
    We order pizzasfrom Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. Expressions
  • 5.
    We order pizzasfrom Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Expressions
  • 6.
    We order pizzasfrom Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions
  • 7.
    We order pizzasfrom Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810.
  • 8.
    We order pizzasfrom Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
  • 9.
    We order pizzasfrom Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions Definition: Mathematical expressions are calculation procedures which are written with numbers, variables, operation symbols +, –, *, / and ( )’s. If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
  • 10.
    We order pizzasfrom Pizza Grande. Each pizza is $8 and there is a $10 delivery charge. Hence if we ordered 5 pizzas delivered, the total cost would be 8(5) + 10 = $50, excluding the tip. If we want x pizzas delivered, then the total cost is given by the formula “8x + 10”. Such a formula is called an expression. Expressions Definition: Mathematical expressions are calculation procedures which are written with numbers, variables, operation symbols +, –, *, / and ( )’s. Expressions calculate the expected future results. If we ordered x = 100 pizzas, the cost would be 8(100)+10 = $810. The value x = 100 is called the input and the projected cost $810 is called the output.
  • 11.
    An algebraic expressionis a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions
  • 12.
    An algebraic expressionis a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Trigonometric or log-formulas are not algebraic.
  • 13.
    An algebraic expressionis a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4,
  • 14.
    An algebraic expressionis a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 ,
  • 15.
    An algebraic expressionis a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 , (x1/2 + y)1/3 (4y2 – (x + 4)1/2)1/4
  • 16.
    An algebraic expressionis a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 , (x1/2 + y)1/3 (4y2 – (x + 4)1/2)1/4 Examples of non–algebraic expressions are sin(x), 2x, log(x + 1).
  • 17.
    An algebraic expressionis a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 , (x1/2 + y)1/3 (4y2 – (x + 4)1/2)1/4 Examples of non–algebraic expressions are sin(x), 2x, log(x + 1). The algebraic expressions anxn + an–1xn–1...+ a1x + a0 where ai are numbers, are called polynomials (in x).
  • 18.
    An algebraic expressionis a formula constructed with variables and numbers using addition, subtraction, multiplication, division, and taking roots. Algebraic Expressions Examples of algebraic expressions are 3x2 – 2x + 4, x2 + 3 3 x3 – 2x – 4 , (x1/2 + y)1/3 (4y2 – (x + 4)1/2)1/4 Examples of non–algebraic expressions are sin(x), 2x, log(x + 1). The algebraic expressions anxn + an–1xn–1...+ a1x + a0 where ai are numbers, are called polynomials (in x). The algebraic expressions where P and Q are polynomials, are called rational expressions. P Q
  • 19.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions.
  • 20.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – (3x – 4)(x + 5)
  • 21.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – (3x – 4)(x + 5) The point of this problem is how to subtract a “product”.
  • 22.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert [ ] The point of this problem is how to subtract a “product”.
  • 23.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] Insert [ ] remove [ ]
  • 24.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 Insert [ ] remove [ ]
  • 25.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] remove [ ]
  • 26.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] remove [ ]
  • 27.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] remove [ ] (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = … Or distribute the minus sign and change it to an addition problem:
  • 28.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] To factor an expression means to write it as a product in a non-obvious way. remove [ ] (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = … Or distribute the minus sign and change it to an addition problem:
  • 29.
    Polynomial Expressions Following areexamples of operations with polynomials and rational expressions. Example A. Expand and simplify. (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 + 11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Insert [ ] A3 B3 = (A B)(A2 AB + B2) Important Factoring Formulas: To factor an expression means to write it as a product in a non-obvious way. A2 – B2 = (A + B)(A – B) +– +– +– remove [ ] (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = … Or distribute the minus sign and change it to an addition problem:
  • 30.
    Example B. Factor64x3 + 125 Polynomial Expressions A3 B3 = (A B)(A2 AB + B2)+– + –+–
  • 31.
    Example B. Factor64x3 + 125 64x3 + 125 = (4x)3 + (5)3 Polynomial Expressions A3 B3 = (A B)(A2 AB + B2)+– + –+– A3 B3
  • 32.
    Example B. Factor64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) Polynomial Expressions A3 B3 = (A B)(A2 AB + B2)+– + –+– (A B) (A2 AB + B2)+ –
  • 33.
    Example B. Factor64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions A3 B3 = (A B)(A2 AB + B2)+– + –+–
  • 34.
    Example B. Factor64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions We factor polynomials for the following purposes. A3 B3 = (A B)(A2 AB + B2)+– + –+–
  • 35.
    Example B. Factor64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions We factor polynomials for the following purposes. I. It’s easier to calculate an output or to check the sign of an output using the factored form. A3 B3 = (A B)(A2 AB + B2)+– + –+–
  • 36.
    Example B. Factor64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions We factor polynomials for the following purposes. I. It’s easier to calculate an output or to check the sign of an output using the factored form. II. To simplify or perform algebraic operations with rational expressions. A3 B3 = (A B)(A2 AB + B2)+– + –+–
  • 37.
    Example B. Factor64x3 + 125 64x3 + 125 = (4x)3 + (5)3 = (4x + 5)((4x)2 – (4x)(5) +(5)2) = (4x + 5)(16x2 – 20x + 25) Polynomial Expressions We factor polynomials for the following purposes. I. It’s easier to calculate an output or to check the sign of an output using the factored form. II. To simplify or perform algebraic operations with rational expressions. III. To solve equations (See next section). A3 B3 = (A B)(A2 AB + B2)+– + –+–
  • 38.
    Evaluate Polynomial Expressions It'seasier to evaluate factored polynomial expressions. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first.
  • 39.
    Evaluate Polynomial Expressions It'seasier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly. Example C. Evaluate 2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first.
  • 40.
    Example C. Evaluate2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
  • 41.
    Example C. Evaluate2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
  • 42.
    Example C. Evaluate2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
  • 43.
    Example C. Evaluate2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40 Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
  • 44.
    Example C. Evaluate2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40 Plug in x = –1: –1 [2(–1) – 1] [(–1) – 2] Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
  • 45.
    Example C. Evaluate2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40 Plug in x = –1: –1 [2(–1) – 1] [(–1) – 2] = –1 [–3] [–3] = –9 Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
  • 46.
    Example C. Evaluate2x3 – 5x2 + 2x for x = –2, –1, 3 by factoring it first. 2x3 – 5x2 + 2x = x(2x2 – 5x + 2) = x(2x – 1)(x – 2) Plug in x = –2: –2 [2(–2) – 1] [(–2) – 2] = –2 [–5] [–4] = –40 Plug in x = –1: –1 [2(–1) – 1] [(–1) – 2] = –1 [–3] [–3] = –9 Plug in x = 3: 3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15 Evaluate Polynomial Expressions It's easier to evaluate factored polynomial expressions. It takes fewer steps then plugging in the values directly.
  • 47.
    Determine the Signsof the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
  • 48.
    Example D. Determinewhether the outcome is + or – for x2 – 2x – 3 if x = –3/2. Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
  • 49.
    Example D. Determinewhether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
  • 50.
    Example D. Determinewhether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
  • 51.
    Example D. Determinewhether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + . Determine the Signs of the Outputs. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
  • 52.
    Example D. Determinewhether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + . Determine the Signs of the Outputs. Rational Expressions We say a rational expression is in the factored form if it's numerator and denominator are factored. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
  • 53.
    Example D. Determinewhether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + . Determine the Signs of the Outputs. Rational Expressions We say a rational expression is in the factored form if it's numerator and denominator are factored. Example E. Factor x2 – 1 x2 – 3x+ 2 It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
  • 54.
    Example D. Determinewhether the outcome is + or – for x2 – 2x – 3 if x = –3/2. x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2, we get (–3/2 – 3)(–3/2 + 1) is (–)(–) = + . Determine the Signs of the Outputs. Rational Expressions We say a rational expression is in the factored form if it's numerator and denominator are factored. Example E. Factor x2 – 1 x2 – 3x+ 2 x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1) (x – 1)(x – 2) is the factored form. It's easier to determine the sign of an output, when evaluating an expression, using the factored form.
  • 55.
    Rational Expressions We putrational expressions in the factored form in order to reduce, multiply or divide them.
  • 56.
    Rational Expressions We putrational expressions in the factored form in order to reduce, multiply or divide them. Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e. x*y x*z = x*y x*z = y z 1
  • 57.
    Rational Expressions We putrational expressions in the factored form in order to reduce, multiply or divide them. x*y x*z = x*y x*z = y z A rational expression that can't be cancelled any further is said to be reduced. Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e.
  • 58.
    Rational Expressions We putrational expressions in the factored form in order to reduce, multiply or divide them. Example F. Reduce x2 – 1 x2 – 3x+ 2 x*y x*z = x*y x*z = y z A rational expression that can't be cancelled any further is said to be reduced. Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e.
  • 59.
    Rational Expressions We putrational expressions in the factored form in order to reduce, multiply or divide them. Example F. Reduce x2 – 1 x2 – 3x+ 2 x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1) (x – 1)(x – 2) x*y x*z = x*y x*z = y z A rational expression that can't be cancelled any further is said to be reduced. factor Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e.
  • 60.
    Rational Expressions We putrational expressions in the factored form in order to reduce, multiply or divide them. Example F. Reduce x2 – 1 x2 – 3x+ 2 x2 – 1 x2 – 3x+ 2 = (x – 1)(x + 1) (x – 1)(x – 2) x*y x*z = x*y x*z = y z A rational expression that can't be cancelled any further is said to be reduced. = (x + 1) (x – 2) factor Cancellation Rule: Given a rational expression in the factored form, common factors may be cancelled, i.e.
  • 61.
  • 62.
  • 63.
    Rational Expressions Multiplication Rule: P Q R S *= P*R Q*S Division Rule: P Q R S ÷
  • 64.
    Rational Expressions Multiplication Rule: P Q R S *= P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate
  • 65.
    Rational Expressions Multiplication Rule: P Q R S *= P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2)
  • 66.
    Rational Expressions Multiplication Rule: P Q R S *= P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* Reciprocate
  • 67.
    Rational Expressions Multiplication Rule: P Q R S *= P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* Reciprocate To carry out these operations, put the expressions in factored form and cancel as much as possible.
  • 68.
    Rational Expressions Multiplication Rule: Tocarry out these operations, put the expressions in factored form and cancel as much as possible. P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* = 2(x – 3) (y + 3) (y + 3)(y – 1) (3 – x)(3 + x)* factor and cancel
  • 69.
    Rational Expressions Multiplication Rule: Tocarry out these operations, put the expressions in factored form and cancel as much as possible. P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* = 2(x – 3) (y + 3) (y + 3)(y – 1) (3 – x)(3 + x)* 1 factor and cancel
  • 70.
    Rational Expressions Multiplication Rule: Tocarry out these operations, put the expressions in factored form and cancel as much as possible. P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* = 2(x – 3) (y + 3) (y + 3)(y – 1) (3 – x)(3 + x)* –1 1 factor and cancel
  • 71.
    Rational Expressions Multiplication Rule: Tocarry out these operations, put the expressions in factored form and cancel as much as possible. P Q R S * = P*R Q*S Division Rule: P Q R S ÷ = P*S Q*R Reciprocate Example G. Simplify (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) (2x – 6) (y + 3) ÷ (y2 + 2y – 3) (9 – x2) = (2x – 6) (y + 3) (y2 + 2y – 3) (9 – x2)* = 2(x – 3) (y + 3) (y + 3)(y – 1) (3 – x)(3 + x)* –1 1 = –2(y – 1) (x + 3)
  • 72.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions
  • 73.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions II. to simplify complex fractions
  • 74.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions II. to simplify complex fractions III. to solve rational equations (later)
  • 75.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 II. to simplify complex fractions III. to solve rational equations (later)
  • 76.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 Combining Rational Expressions (LCD Method): II. to simplify complex fractions III. to solve rational equations (later)
  • 77.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, II. to simplify complex fractions III. to solve rational equations (later)
  • 78.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, II. to simplify complex fractions III. to solve rational equations (later)
  • 79.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 7 12 5 8 + – 16 9 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, 48 II. to simplify complex fractions III. to solve rational equations (later)
  • 80.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 7 12 5 8 + – 16 9 48 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. II. to simplify complex fractions III. to solve rational equations (later)
  • 81.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 7 12 5 8 + – 16 94 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 II. to simplify complex fractions III. to solve rational equations (later)
  • 82.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 67 12 5 8 + – 16 94 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 II. to simplify complex fractions III. to solve rational equations (later)
  • 83.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 67 12 5 8 + – 16 94 3 Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 II. to simplify complex fractions III. to solve rational equations (later)
  • 84.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 67 12 5 8 + – 16 94 3 = Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 28 + 30 – 27 48 II. to simplify complex fractions III. to solve rational equations (later)
  • 85.
    Rational Expressions The leastcommon denominator (LCD) is needed I. to combine (add or subtract) rational expressions Example H: Combine 7 12 5 8 + – 16 9 The LCD = 48, ( )* 48 67 12 5 8 + – 16 94 3 = Combining Rational Expressions (LCD Method): To combine rational expressions (F ± G), multiple (F ± G)* LCD/LCD, expand (F ± G)* LCD and simplify (F ± G)(LCD) / LCD. 48 28 + 30 – 27 48 = 48 31 II. to simplify complex fractions III. to solve rational equations (later)
  • 86.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3Example I. Combine
  • 87.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Example I. Combine
  • 88.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), Example I. Combine
  • 89.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, Example I. Combine
  • 90.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y2 + 2y – 3)(y – 1)(y + 2) 2y – 1 y – 3[ ] Example I. Combine
  • 91.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y2 + 2y – 3)(y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) Example I. Combine LCD
  • 92.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) Example I. Combine LCD expand and simplify. (y – 1)(y + 3)
  • 93.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) (y + 3) Example I. Combine LCD expand and simplify. (y – 1)(y + 3)
  • 94.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) (y + 3) (y + 2) Example I. Combine LCD expand and simplify. (y – 1)(y + 3)
  • 95.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) = (2y – 1)(y + 3) – (y – 3)(y + 2) (y + 3) (y + 2) Example I. Combine LCD LCD expand and simplify. (y – 1)(y + 3)
  • 96.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) = (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3 (y + 3) (y + 2) Example I. Combine LCD LCDLCD expand and simplify. (y – 1)(y + 3)
  • 97.
    Rational Expressions – (y2 +2y – 3)(y2 + y – 2) 2y – 1 y – 3 y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3) Hence the LCD = (y – 1)(y + 2)(y + 3), multiplying LCD/LCD (= 1) to the problem, – (y – 1)(y + 2) 2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3) = (2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3 So – (y2 + 2y – 3)(y2 + y – 2) 2y – 1 y – 3 = y2 + 6y + 3 (y – 1)(y + 2)(y + 3) (y + 3) (y + 2) Example I. Combine LCD LCDLCD expand and simplify. (y – 1)(y + 3)
  • 98.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. 2 3 –4 1 3 2
  • 99.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 –4 1 3 2
  • 100.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Their LCD is 12.
  • 101.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 Their LCD is 12.
  • 102.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 Their LCD is 12. = 1
  • 103.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12
  • 104.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4
  • 105.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4 6
  • 106.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4 6 3 4
  • 107.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4 6 3 4 = 3 4 – 18 – 8
  • 108.
    Rational Expressions Example J.Simplify –3 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. 2 3 The fractional terms are –4 1 3 2 3 1 2 3 4 1 3 2 .,,, Multiplying 12/12 (or 1) to the problem: –3 1 2 3 –4 1 3 2 ( ) )( 12 12 = Their LCD is 12. –3 1 2 3 –4 1 3 2 *12 *12 *12 *12 4 6 3 4 = 3 4 – 18 – 8 = 14 5
  • 109.
    Rational Expressions Example K.Simplify – (x – h) 1 A complex fraction is a fraction of fractions. To simplify a complex fraction, use the LCD to clear all the denominators of all the fractioned terms. (x + h) 1 2h Multiply the top and bottom by (x – h)(x + h) to reduce the expression in the numerators to polynomials. – (x – h) 1 (x + h) 1 2h = – (x – h) 1 (x + h) 1 2h (x + h)(x – h)[ ] (x + h)(x – h)* = –(x + h) (x – h) 2h(x + h)(x – h) = 2h 2h(x + h)(x – h) = 1 (x + h)(x – h)
  • 110.
    To rationalize radicalsin expressions we often use the formula (x – y)(x + y) = x2 – y2. Rationalize Radicals
  • 111.
    To rationalize radicalsin expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals
  • 112.
    Example K: Rationalizethe numerator To rationalize radicals in expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. h x + h – x Rationalize Radicals
  • 113.
    To rationalize radicalsin expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * Example K: Rationalize the numerator h x + h – x
  • 114.
    To rationalize radicalsin expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * = h (x + h)2 – (x)2 (x + h + x) Example K: Rationalize the numerator h x + h – x
  • 115.
    To rationalize radicalsin expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * = h (x + h)2 – (x)2 (x + h + x) Example K: Rationalize the numerator h x + h – x (x + h) – (x) = h
  • 116.
    To rationalize radicalsin expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * = h (x + h)2 – (x)2 (x + h + x) = h h (x + h + x) Example K: Rationalize the numerator h x + h – x (x + h) – (x) = h
  • 117.
    To rationalize radicalsin expressions we often use the formula (x – y)(x + y) = x2 – y2. (x + y) and (x – y) are called conjugates. Rationalize Radicals h x + h – x = h (x + h – x) (x + h + x) (x + h + x) * = h (x + h)2 – (x)2 (x + h + x) = h h (x + h + x) = 1 x + h + x Example K: Rationalize the numerator h x + h – x (x + h) – (x) = h
  • 118.
    Exercise A. Factoreach expression then use the factored form to evaluate the given input values. No calculator. Applications of Factoring 1. x2 – 3x – 4, x = –2, 3, 5 2. x2 – 2x – 15, x = –1, 4, 7 3. x2 – x – 2, x = ½ ,–2, –½ 4. x3 – 2x2, x = –2, 2, 4 5. x4 – 3x2, x = –1, 1, 5 6. x3 – 4x2 – 5x, x = –4, 2, 6 B. Determine if the output is positive or negative using the factored form. 7. x2 – 4 x + 4 8. x3 – 2x2 x2 – 2x + 1 , x = –3, 1, 5 , x = –0.1, 1/2, 5 4. x2 – 4 x + 4 5. x2 + 2x – 3 x2 + x 6. x3 – 2x2 x2 – 2x + 1 , x = –3.1, 1.9 , x = –0.1, 0.9, 1.05 , x = –0.1, 0.99, 1.01 1. x2 – 3x – 4, x = –2½, –2/3, 2½, 5¼ 2. –x2 + 2x + 8, x = –2½, –2/3, 2½, 5¼ 3. x3 – 2x2 – 8x, x = –4½, –3/4, ¼, 6¼,
  • 119.
    C. Simplify. Donot expand the results. Multiplication and Division of Rational Expressions 1. 10x * 2 5x3 15x 4 * 16 25x42. 3.12x6 * 5 6x14 4. 75x 49 * 42 25x3 5. 2x – 4 2x + 4 5x + 10 3x – 6 6. x + 4 –x – 4 4 – x x – 4 7. 3x – 9 15x – 5 3 – x 5 – 15x 8. 42 – 6x –2x + 14 4 – 2x –7x + 14 * * * * 9. (x2 – x – 2 ) (x2 – 1) (x2 + 2x + 1) (x2 + x ) * 10. (x2 + 5x – 6 ) (x2 + 5x + 6) (x2 – 5x – 6 ) (x2 – 5x + 6) * 11. (x2 – 3x – 4 ) (x2 – 1) (x2 – 2x – 8) (x2 – 3x + 2) * 12. (– x2 + 6 – x ) (x2 + 5x + 6) (x2 – x – 12) (6 – x2 – x) * 13.(3x2 – x – 2) (x2 – x + 2) (3x2 + 4x + 1) (–x – 3x2) 14. (x + 1 – 6x2) (–x2 – 4) (2x2 + x – 1 ) (x2 – 5x – 6) 15. (x3 – 4x) (–x2 + 4x – 4) (x2 + 2) (–x + 2) 16. (–x3 + 9x ) (x2 + 6x + 9) (x2 + 3x) (–3x2 – 9x) ÷ ÷ ÷ ÷
  • 120.
    Multiplication and Divisionof Rational Expressions D. Multiply, expand and simplify the results. 1. x + 3 x + 1 (x2 – 1) 2. x – 3 x – 2 (x2 – 4) 3. 2x + 3 1 – x (x2 – 1) 4. 3 – 2x x + 2 (x + 2)(x +1) 5. 3 – 2x 2x – 1 (3x + 2)(1 – 2x) 6. x – 2 x – 3 ( x + 1 x + 3)( x – 3)(x + 3) 7. 2x – 1 x + 2 ( – x + 2 2x – 3 )( 2x – 3)(x + 2) + 8. x – 2 x – 3 ( x + 1 x + 3 ) ( x – 3)(x + 3)– 9. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) 10. x + 3 x2 – 4 ( – 2x + 1 x2 + x – 2 ) ( x – 2)(x + 2)(x – 1) 11. x – 1 x2 – x – 6 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 2)(x + 1)
  • 121.
    E. Combine andsimplify the answers. –3 x – 3 + 2x –6 – 2x 3. 2x – 3 x – 3 – 5x + 4 5 – 15x 4. 3x + 1 6x – 4 – 2x + 3 2 – 3x5. –5x + 7 3x – 12+ 4x – 3 –2x + 86. 3x + 1 + x + 3 4 – x211. x2 – 4x + 4 x – 4 + x + 5 –x2 + x + 2 12. x2 – x – 6 3x + 1 + 2x + 3 9 – x213. x2 – x – 6 3x – 4 – 2x + 5 x2 – x – 6 14. –x2 + 5x + 6 3x + 4 + 2x – 3 –x2 – 2x + 3 15. x2 – x 5x – 4 – 3x – 5 1 – x216. x2 + 2x – 3 –3 2x – 1 + 2x 2 – 4x 1. 2x – 3 x – 2 + 3x + 4 5 – 10x 2. 3x + 1 2x – 5 – 2x + 3 5 – 10x 9. –3x + 2 3x – 12 + 7x – 2 –2x + 8 10. 3x + 5 3x –2 – x + 3 2 – 3x7. –5x + 7 3x – 4 + 4x – 3 –6x + 88. Addition and Subtraction of Fractions
  • 122.
    Complex Fractions 1 2x +1 – 2 3 – 1 2x + 1 3. –2 2x + 1 – + 3 x + 4 4. 1 x + 4 2 2x + 1 4 2x + 3 – + 3 x + 4 5. 3 3x – 2 5 3x – 2 –5 2x + 5 – + 3 –x + 4 6. 2 2x – 3 6 2x – 3 2 3 + 2 2 – – 1 6 2 3 1 2+ 1. 1 2 – + 5 6 2 3 1 4 – 2. 3 4 3 2 + F. Combine and simplify the answers. 7. 2 x – 1 – + 3 x + 3 x x + 3 x x – 1 8. 3 x + 2 – + 3 x + 2 x x – 2 x x – 2 9. 2 x + h – 2 x h 10. 3 x – h – 3 x h 11. 2 x + h – 2 x – h h 12. 3 x + h – h 3 x – h
  • 123.
    G. Rationalize thedenominator. 1. 1 – 3 1 + 3 2. 5 + 2 3 – 2 3. 1 – 33 2 + 3 4. 1 – 53 4 + 23 5. 32 – 33 22 – 43 6. 25 + 22 34 – 32 7. 42 – 37 22 – 27 8. x + 3 x – 3 9. 3x – 3 3x + 2 10. x – 2 x + 2 + 2 11. x – 4 x – 3 – 1 Algebra of Radicals