Algebra Presentation on Topic Modulus Function and Polynomials
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Algebra Presentation on Topic Modulus Function and Polynomials Created by: Tasha and Sheryl Performance Task Binus School Bekasi
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polynomial
1) The document discusses polynomials and their properties. It defines what a polynomial is - an expression with terms containing variables and coefficients.
2) It introduces the concept of the degree of a polynomial and defines it as the highest power of the variable terms. Constant polynomials have a degree of zero.
3) The Remainder Theorem is discussed, stating that when dividing a polynomial p(x) by a linear polynomial x-a, the remainder is equal to the value of p(a).
Up1 math t 2012
The document is a mathematics test for a pre-university class containing two sections - Section A with 6 multiple choice questions worth a total of 45 marks, and Section B with two extended response questions worth 15 marks each. Question 1 in Section A asks students to express a fraction in partial fractions, question 2 asks them to express a trigonometric expression in an alternative form and solve an equation, and subsequent questions cover logarithms, inequalities, trigonometric substitutions, and factorizing polynomials. Section B offers two word problems, one involving composite functions and another involving factorizing a polynomial based on given information.
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6.4 factoring and solving polynomial equations
The document provides examples and instructions for factoring polynomials of various types, including:
- Trinomials like x^2 - 5x - 12
- Sum and difference of cubes like x^3 + 8 and 8x^3 - 1
- Polynomials with a common monomial factor like 6x^2 + 15x
- Quadratics in the form of au^2 + bu + c
It also discusses using the zero product property to solve polynomial equations by factoring and setting each factor equal to zero.
Algebra Presentation on Topic Modulus Function and Polynomials
Algebra Presentation on Topic Modulus Function and PolynomialsCreated by : [Dona, Brian]Performance Task Binus School Bekasi
Polynomials
This document provides an overview of polynomials, including:
- Defining polynomials as expressions involving variables and coefficients using addition, subtraction, multiplication, and exponents.
- Discussing the history of polynomial notation pioneered by Descartes.
- Explaining the different types of polynomials like monomials, binomials, and trinomials.
- Outlining common uses of polynomials in mathematics, science, and other fields.
- Describing how to find the degree of a polynomial and graph polynomial functions.
- Explaining arithmetic operations like addition, subtraction, and division that can be performed on polynomials.
Division Of Polynomials
Division of polynomials follows the same rules as division of real numbers. To divide a polynomial by a monomial, each term of the dividend is divided by the monomial divisor. To divide a polynomial by a polynomial, long division is used by repeatedly determining the quotient and remainder until the division is complete. The steps of long division are shown through an example of dividing a third degree polynomial by a linear polynomial divisor. The quotient and remainder are checked by multiplying the divisor and quotient together.
Solving polynomial equations in factored form
The document provides instructions for solving polynomial equations in factored form. It begins by explaining that to solve an equation like (x – 5)(x + 4) = 0, one should not use the FOIL method but rather split the equation into two separate problems that each equal zero: x – 5 = 0 and x + 4 = 0. It then works through several examples of solving factored polynomial equations by finding the values of x that make each factor equal to zero. The document also covers factoring out the greatest common factor from expressions.
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1) The document discusses polynomials and their properties. It defines what a polynomial is - an expression with terms containing variables and coefficients.
2) It introduces the concept of the degree of a polynomial and defines it as the highest power of the variable terms. Constant polynomials have a degree of zero.
3) The Remainder Theorem is discussed, stating that when dividing a polynomial p(x) by a linear polynomial x-a, the remainder is equal to the value of p(a).
Up1 math t 2012
The document is a mathematics test for a pre-university class containing two sections - Section A with 6 multiple choice questions worth a total of 45 marks, and Section B with two extended response questions worth 15 marks each. Question 1 in Section A asks students to express a fraction in partial fractions, question 2 asks them to express a trigonometric expression in an alternative form and solve an equation, and subsequent questions cover logarithms, inequalities, trigonometric substitutions, and factorizing polynomials. Section B offers two word problems, one involving composite functions and another involving factorizing a polynomial based on given information.
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Steps are shown for solving systems of equations by the elimination method, including multiplying one equation by a constant to eliminate a variable before adding the equations. Examples are provided of solving systems by substitution, where one variable is solved for in one equation and substituted into the other equation. Practice problems are given for students to apply
6.4 factoring and solving polynomial equations
The document provides examples and instructions for factoring polynomials of various types, including:
- Trinomials like x^2 - 5x - 12
- Sum and difference of cubes like x^3 + 8 and 8x^3 - 1
- Polynomials with a common monomial factor like 6x^2 + 15x
- Quadratics in the form of au^2 + bu + c
It also discusses using the zero product property to solve polynomial equations by factoring and setting each factor equal to zero.
Algebra Presentation on Topic Modulus Function and Polynomials
Algebra Presentation on Topic Modulus Function and PolynomialsCreated by : [Dona, Brian]Performance Task Binus School Bekasi
Polynomials
This document provides an overview of polynomials, including:
- Defining polynomials as expressions involving variables and coefficients using addition, subtraction, multiplication, and exponents.
- Discussing the history of polynomial notation pioneered by Descartes.
- Explaining the different types of polynomials like monomials, binomials, and trinomials.
- Outlining common uses of polynomials in mathematics, science, and other fields.
- Describing how to find the degree of a polynomial and graph polynomial functions.
- Explaining arithmetic operations like addition, subtraction, and division that can be performed on polynomials.
Division Of Polynomials
Division of polynomials follows the same rules as division of real numbers. To divide a polynomial by a monomial, each term of the dividend is divided by the monomial divisor. To divide a polynomial by a polynomial, long division is used by repeatedly determining the quotient and remainder until the division is complete. The steps of long division are shown through an example of dividing a third degree polynomial by a linear polynomial divisor. The quotient and remainder are checked by multiplying the divisor and quotient together.
Solving polynomial equations in factored form
The document provides instructions for solving polynomial equations in factored form. It begins by explaining that to solve an equation like (x – 5)(x + 4) = 0, one should not use the FOIL method but rather split the equation into two separate problems that each equal zero: x – 5 = 0 and x + 4 = 0. It then works through several examples of solving factored polynomial equations by finding the values of x that make each factor equal to zero. The document also covers factoring out the greatest common factor from expressions.
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Here are the steps to solve the quadratic equations using the quadratic formula:
1) Write the equation in standard form: ax^2 + bx + c = 0
2) Identify the coefficients a, b, c
3) Plug the coefficients into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
4) Simplify the solution
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- Terms are numbers or products of numbers and variables raised to powers. Coefficients are numerical factors of terms. Constants are terms that are only numbers.
- Polynomials are sums of terms involving variables raised to whole number exponents, with no variables in denominators.
- Types of polynomials include monomials (1 term), binomials (2 terms), and trinomials (3 terms). Degree is the largest exponent of any term.
- Operations on polynomials include adding/subtracting like terms, multiplying using distribution and FOIL, dividing using long division, and special products like (a+b)2 and (a+b)(a
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To summarize the key steps for factoring polynomials:
1. Determine possible integer roots by finding the divisors of the constant term.
2. Use the remainder theorem or Ruffini's rule to check if an integer is a root by dividing the polynomial by (x - a) and checking if the remainder is zero.
3. Integer roots that produce a zero remainder are factors of the polynomial. Repeating this process allows one to fully factor the polynomial into linear terms.
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Here are the steps to solve the quadratic equations using the quadratic formula:
1) Write the equation in standard form: ax^2 + bx + c = 0
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x = (-b ± √(b^2 - 4ac)) / 2a
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1) Summing algebraic expressions by adding like terms.
2) Subtracting algebraic expressions by adding the opposite of like terms.
3) Multiplying algebraic expressions by multiplying each term in one factor by each term in the other factor.
4) Dividing algebraic expressions using long division.
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The document discusses polynomials and polynomial functions. It defines a polynomial as a sum of monomials, with a monomial being a variable or the product of a variable and real numbers with whole number exponents. It classifies polynomials by degree and number of terms, with examples of common types like linear, quadratic, and cubic polynomials. It also defines a polynomial function as a function represented by a polynomial, and discusses finding sums, differences, and writing polynomials in standard form.
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The document defines key polynomial vocabulary including:
- Terms are numbers or products of numbers and variables raised to powers. Coefficients are numerical factors of terms. Constants are terms that are only numbers.
- Polynomials are sums of terms involving variables raised to whole number exponents, with no variables in denominators.
- Types of polynomials include monomials (1 term), binomials (2 terms), and trinomials (3 terms). Degree is the largest exponent of any term.
- Operations on polynomials include adding/subtracting like terms, multiplying using distribution and FOIL, dividing using long division, and special products like (a+b)2 and (a+b)(a
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1. Determine possible integer roots by finding the divisors of the constant term.
2. Use the remainder theorem or Ruffini's rule to check if an integer is a root by dividing the polynomial by (x - a) and checking if the remainder is zero.
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1) Find the value of p given one root of an equation.
2) Solve example quadratic equations and express a constant h in terms of k.
3) Determine possible values of p if an equation has two equal roots.
4) Solve another example quadratic equation to three decimal places.
The answers provided are: p = 5, x = 1/3 or x = -2, p = 8 or p
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Algebra Presentation on Topic Modulus Function and Polynomials
- 1. Performance Task on Algebra By: Tasha & Sheryl
- 3. Question
- 4. Answer
- 6. Question The polynomial x3–4x2+ax+b, where a and b are constants, is denoted by f(x). when f(x) is divided by (x-2) the remainder is 6 and when f(x) is divided by (x+1) the remainder is 3. Find the value of a and b!
- 7. Answer
- 8. Thank You!!