(1) The document provides an overview of intermediate algebra topics including exponents, factoring, polynomials, and solving equations. (2) Key concepts discussed include integer exponents, the product rule, factoring common factors, special factoring patterns, and solving polynomial equations by factoring and applying the zero-factor theorem. (3) The goal is to review and reinforce these essential intermediate algebra skills.
OBJECTIVES
Revision On:
Simplify of Algebraic Fraction
Perform Operations on Algebraic Fraction
Solve Equations Involving Algebraic Fraction
Make Substitution in Algebraic Fraction
Solve Simultaneous Equations Involving Algebraic Fraction
Undefined value of an Algebraic Fraction
Represent Algebraic Fractions Graphically.
DEFINITION OF A QUADRATIC EQUATION A quadratic equatioLinaCovington707
DEFINITION OF A QUADRATIC EQUATION
A quadratic equation in x is an equation that can be written in the general form
ax bx c2 0,
where a, b, and c are real numbers, with a ≠ 0.
A quadratic equation in x is also called a second-degree polynomial equation in x.
THE ZERO-PRODUCT PRINCIPLE
To solve a quadratic equation by factoring, we apply the zero-product principle which states that:
If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.
If AB = 0, then A = 0 or B = 0.
SOLVING A QUADRATIC EQUATION BY FACTORING
2. Factor completely.
3. Apply the zero-product principle, setting each factor containing a variable equal to zero.
4. Solve the equations in step 3.
5. Check the solutions in the original equation.
EXAMPLE: SOLVING QUADRATIC
EQUATIONS BY FACTORING (1 OF 3)
Solve by factoring: 2x2 x 1.
Step 1 Move all nonzero terms to one side and obtain zero on the other side.
2x x2 1 0
Step 2 Factor
(2x− 1)(x + 1) = 0
EQUATIONS BY FACTORING (2 OF 3)
Steps 3 and 4 Set each factor equal to zero and solve the resulting equations.
(2x− 1)(x + 1) = 0
EXAMPLE: SOLVING QUADRATIC
EQUATIONS BY FACTORING (3 OF 3)
Step 5 Check the solutions in the original equation.
2x x2 1
Check
EXAMPLE: SOLVING QUADRATIC
EXAMPLE: SOLVING QUADRATIC EQUATIONS BY FACTORING
•
Solution
• Step 1- Move all nonzero terms to one side and obtain zero on the other.
• Step 2 Factor- 2x(2x-1)=0
• Step 3 and 4- Set each factor equal to zero and solve the resulting equations.
2x=0 or 2x-1=0 x=0
2x=1
x=1/2
• Step 5
B) SOLUTION
CHECK POINT SOLVE BY FACTORING
SOLVING QUADRATIC EQUATIONS BY THE SQUARE ROOT PROPERTY
EXAMPLE: SOLVING QUADRATIC EQUATIONS BY THE SQUARE ROOT PROPERTY
Solve by the square root property:
SOLUTION
SOLUTION
SOLUTION
CHECK POINT: SOLVE BY THE SQUARE ROOT PROPERTY:
COMPLETING THE SQUARE
EXAMPLE: CREATING PERFECT SQUARE TRINOMIALS BY COMPLETING THE SQUARE
CHECK POINT: SOLVING A QUADRATIC EQUATION BY COMPLETING THE SQUARE
EXAMPLE: SOLVING A QUADRATIC EQUATION BY COMPLETING THE SQUARE
CHECK POINT: SOLVE BY COMPLETING THE SQUARE.
THE QUADRATIC FORMULA
EXAMPLE: SOLVING A QUADRATIC EQUATION USING THE QUADRATIC
FORMULA (1 OF 2)
Solve using the quadratic formula:
2x2 2x 1 0
a = 2, b = 2, c = −1
EXAMPLE: SOLVING A QUADRATIC EQUATION USING THE QUADRATIC FORMULA (2 OF 2)
CHECK POINT:
THE DISCRIMINANT
The discriminant of the quadratic equation determines the number and type of solutions.
THE DISCRIMINANT AND THE KINDS OF SOLUTIONS TO A X SQUARED + B X + C = 0
• If the discriminant is positive, there will be two unequal real solutions.
• If the discriminant is zero, there is one real (repeated) solution. ...
OBJECTIVES
Revision On:
Simplify of Algebraic Fraction
Perform Operations on Algebraic Fraction
Solve Equations Involving Algebraic Fraction
Make Substitution in Algebraic Fraction
Solve Simultaneous Equations Involving Algebraic Fraction
Undefined value of an Algebraic Fraction
Represent Algebraic Fractions Graphically.
DEFINITION OF A QUADRATIC EQUATION A quadratic equatioLinaCovington707
DEFINITION OF A QUADRATIC EQUATION
A quadratic equation in x is an equation that can be written in the general form
ax bx c2 0,
where a, b, and c are real numbers, with a ≠ 0.
A quadratic equation in x is also called a second-degree polynomial equation in x.
THE ZERO-PRODUCT PRINCIPLE
To solve a quadratic equation by factoring, we apply the zero-product principle which states that:
If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.
If AB = 0, then A = 0 or B = 0.
SOLVING A QUADRATIC EQUATION BY FACTORING
2. Factor completely.
3. Apply the zero-product principle, setting each factor containing a variable equal to zero.
4. Solve the equations in step 3.
5. Check the solutions in the original equation.
EXAMPLE: SOLVING QUADRATIC
EQUATIONS BY FACTORING (1 OF 3)
Solve by factoring: 2x2 x 1.
Step 1 Move all nonzero terms to one side and obtain zero on the other side.
2x x2 1 0
Step 2 Factor
(2x− 1)(x + 1) = 0
EQUATIONS BY FACTORING (2 OF 3)
Steps 3 and 4 Set each factor equal to zero and solve the resulting equations.
(2x− 1)(x + 1) = 0
EXAMPLE: SOLVING QUADRATIC
EQUATIONS BY FACTORING (3 OF 3)
Step 5 Check the solutions in the original equation.
2x x2 1
Check
EXAMPLE: SOLVING QUADRATIC
EXAMPLE: SOLVING QUADRATIC EQUATIONS BY FACTORING
•
Solution
• Step 1- Move all nonzero terms to one side and obtain zero on the other.
• Step 2 Factor- 2x(2x-1)=0
• Step 3 and 4- Set each factor equal to zero and solve the resulting equations.
2x=0 or 2x-1=0 x=0
2x=1
x=1/2
• Step 5
B) SOLUTION
CHECK POINT SOLVE BY FACTORING
SOLVING QUADRATIC EQUATIONS BY THE SQUARE ROOT PROPERTY
EXAMPLE: SOLVING QUADRATIC EQUATIONS BY THE SQUARE ROOT PROPERTY
Solve by the square root property:
SOLUTION
SOLUTION
SOLUTION
CHECK POINT: SOLVE BY THE SQUARE ROOT PROPERTY:
COMPLETING THE SQUARE
EXAMPLE: CREATING PERFECT SQUARE TRINOMIALS BY COMPLETING THE SQUARE
CHECK POINT: SOLVING A QUADRATIC EQUATION BY COMPLETING THE SQUARE
EXAMPLE: SOLVING A QUADRATIC EQUATION BY COMPLETING THE SQUARE
CHECK POINT: SOLVE BY COMPLETING THE SQUARE.
THE QUADRATIC FORMULA
EXAMPLE: SOLVING A QUADRATIC EQUATION USING THE QUADRATIC
FORMULA (1 OF 2)
Solve using the quadratic formula:
2x2 2x 1 0
a = 2, b = 2, c = −1
EXAMPLE: SOLVING A QUADRATIC EQUATION USING THE QUADRATIC FORMULA (2 OF 2)
CHECK POINT:
THE DISCRIMINANT
The discriminant of the quadratic equation determines the number and type of solutions.
THE DISCRIMINANT AND THE KINDS OF SOLUTIONS TO A X SQUARED + B X + C = 0
• If the discriminant is positive, there will be two unequal real solutions.
• If the discriminant is zero, there is one real (repeated) solution. ...
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A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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2. 1.1 – Integer Exponents
• For any real number b and any natural
number n, the nth power of b is found by
multiplying b as a factor n times.
n
b b b b b
3. Exponential Expression – an
expression that involves
exponents
• Base – the number being multiplied
• Exponent – the number of factors of the
base.
14. Def: Monomial
• An expression that is a constant
or a product of a constant and
variables that are raised to
whole –number powers.
• Ex: 4x 1.6 2xyz
15. Definitions:
• Coefficient: The numerical
factor in a monomial
• Degree of a Monomial: The
sum of the exponents of all
variables in the monomial.
23. To add or subtract Polynomials
• Combine Like Terms
• May be done with columns or
horizontally
• When subtracting- change the
sign and add
24. Evaluate Polynomial Functions
• Use functional notation to
give a polynomial a name
such as p or q and use
functional notation such as
p(x)
• Can use Calculator
25. Calculator Methods
• 1. Plug In
• 2. Use [Table]
• 3. Use program EVALUATE
• 4. Use [STO->]
• 5. Use [VARS] [Y=]
• 6. Use graph- [CAL][Value]
29. Procedure: Multiply a
polynomial by a monomial
• Use the distributive property to
multiply each term in the
polynomial by the monomial.
• Helpful to multiply the
coefficients first, then the
variables in alphabetical order.
32. Procedure: Multiplying
Polynomials
• 1. Multiply every term in the
first polynomial by every term
in the second polynomial.
• 2. Combine like terms.
• 3. Can be done horizontally or
vertically.
43. Procedure: Determine greatest common
factor GCF of 2 or more monomials
• 1. Determine GCF of numerical
coefficients.
• 2. Determine the smallest
exponent of each exponential
factor whose base is common to
the monomials. Write base with
that exponent.
• 3. Product of 1 and 2 is GCF
44. Factoring Common Factor
• 1. Find the GCF of the terms
• 2. Factor each term with the
GCF as one factor.
• 3. Apply distributive property
to factor the polynomial
46. Factoring when first terms is
negative
• Prefer the first term inside parentheses to be
positive. Factor out the negative of the
GCF.
3
2
20 36
4 (5 9)
xy y
y xy
47. Factoring when GCF is a
polynomial
( 5) ( 5)
( 5)( )
a c b c
c a b
48. Factoring by Grouping – 4 terms
• 1. Check for a common factor
• 2. Group the terms so each group has a
common factor.
• 3. Factor out the GCF in each group.
• 4. Factor out the common binomial factor –
if none , rearrange polynomial
• 5. Check
49. Example – factor by grouping
2 2
32 48 20 30
xy xy y y
2 16 24 10 15
y xy x y
2 2 3 8 5
y y x
50. Ralph Waldo Emerson – U.S.
essayist, poet, philosopher
•“We live in
succession , in
division, in parts, in
particles.”
61. Factoring
• 1. Find two numbers with a product equal
to c and a sum equal to b.
• The factored trinomial will have the form(x
+ ___ ) (x + ___ )
• Where the second terms are the numbers
found in step 1.
• Factors could be combinations of positive
or negative
2
x bx c
62. Factoring
Trial and Error
• 1. Look for a common factor
• 2. Determine a pair of coefficients of first
terms whose product is a
• 3. Determine a pair of last terms whose
product is c
• 4. Verify that the sum of factors yields b
• 5. Check with FOIL Redo
2
ax bx c
63. Factoring ac method
• 1. Determine common factor if any
• 2. Find two factors of ac whose sum is b
• 3. Write a 4-term polynomial in which by
is written as the sum of two like terms
whose coefficients are two factors
determined.
• 4. Factor by grouping.
2
ax bx c
64. Example of ac method
2
6 11 4
x x
2
6 3 8 4
x x x
3 (2 1) 4(2 1)
x x x
(2 1)(3 4)
x x
65. Example of ac method
2 2
5 (8 10 3)
y y y
2 2
5 8 2 12 3
y y y y
2
5 2 4 1 3 4 1
y y y y
2
5 4 1 2 3
y y y
66. Factoring - overview
• 1. Common Factor
• 2. 4 terms – factor by grouping
• 3. 3 terms – possible perfect square
• 4. 2 terms –difference of squares
• Sum of cubes
• Difference of cubes
• Check each term to see if completely
factored
67. Isiah Thomas:
• “I’ve always believed no
matter how many shots I
miss, I’m going to make
the next one.”
70. Example of zero factor property
5 2 0
5 0 2 0
5 2
5,2 2, 5
x x
x or x
x or x
or
71. Solving a polynomial equation by
factoring.
1. Factor the polynomial
completely.
2. Set each factor equal to 0
3. Solve each of resulting equations
4. Check solutions in original
equation.
5. Write the equation in standard
form.
72. Example – solve by factoring
2
3 11 4
x x
2
3 11 4 0
x x
3 1 4 0
x x
3 1 0 4 0
x or x
1
4
3
x or x
73. Example: solve by factoring
3 2
3 2
2
4 12
4 12 0
4 12 0
6 2 0
0,6, 2
x x x
x x x
x x x
x x x
74. Example: solve by factoring
• A right triangle has a
hypotenuse 9 ft longer than the
base and another side 1 foot
longer than the base. How long
are the sides?
• Hint: Draw a picture
• Use the Pythagorean theorem
75. Solution
• Answer: 20 ft, 21 ft, and 29 ft
2 2
2
1 9
x x x
20 4
x or x
76. Example – solve by factoring
• Answer: {-1/2,4}
3 2 7 12
x x
77. Example: solve by factoring
• Answer: {-5/2,2}
2 2
1 1 1
3 2
2 12 3
x x x
78. Example: solve by factoring
• Answer: {0,4/3}
2
9 1 4 6 1 3
y y y y y
79. Example: solve by factoring
• Answer: {-3,-2,2}
3 2
3 13 7 3 1
t t t t
80. Sugar Ray Robinson
• “I’ve always believed that
you can think positive just
as well as you can think
negative.”
81.
82. Maya Angelou - poet
• “Since time is the one
immaterial object which we
cannot influence – neither
speed up nor slow down, add
to nor diminish – it is an
imponderably valuable gift.”