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This document discusses factoring polynomials that are the difference of two squares using the formula a2 - b2 = (a + b)(a - b). It provides examples of factoring polynomials like x2 - 16, 9x2 - 100, and 36m2 - 49n4. It also lists 10 practice problems for factoring polynomials that are differences of two squares and references where readers can learn more.

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rational equation transformable to quadratic equation.pptx

1. The document provides examples for solving quadratic equations that are not in standard form by transforming them into standard form ax2 + bx + c = 0 and then using methods like factoring or the quadratic formula.
2. It also gives examples for solving rational algebraic equations by multiplying both sides by the least common denominator to obtain a quadratic equation, transforming it into standard form, and then solving.
3. The examples cover topics like solving for the solution set, checking solutions, and using the quadratic formula to solve transformed equations.

Solving Quadratic Equations

Provides example on how to solve quadratic equations using extracting square roots, factoring, completing the square and quadratic formula.

55 addition and subtraction of rational expressions

The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.

Polynomial equations

This document discusses finding the real roots of polynomial equations. It states that a polynomial of degree n can have at most n real roots. It then provides examples of factorizing polynomials into their linear factors to find their roots. Finally, it lists 5 polynomial equations and asks the reader to find the roots of each by factorizing.

Synthetic division

This document provides instructions for using synthetic division to divide polynomials. It contains the following key points:
1. Synthetic division can be used to divide polynomials when the divisor has a leading coefficient of 1 and there is a coefficient for every power of the variable in the numerator.
2. The procedure involves writing the terms of the numerator in descending order, bringing down the constant of the divisor, multiplying and adding down the columns to obtain the coefficients of the quotient polynomial and the remainder.
3. An example problem walks through each step of synthetic division to divide (5x^4 - 4x^2 + x + 6) / (x - 3), obtaining a quotient of 5x^3 + 15

7.7 Solving Radical Equations

This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.

Zeros of a polynomial function

The document discusses polynomial functions and their roots. It begins by defining that the roots of a polynomial function are the values of x that make the function equal to 0. It then provides examples of finding the roots of linear and quadratic equations. Next, it introduces the Rational Root Theorem, which states that possible rational roots must be factors of the constant term and leading coefficient. Examples are given to demonstrate applying the theorem. The document concludes by using synthetic division to find all three roots of a cubic polynomial given one known root.

Graphing Quadratics

This document provides information about graphing quadratic functions. It defines the standard form of a quadratic function as y = ax^2 + bx + c and explains that the sign of a determines if the parabola opens up or down. It describes how to find the axis of symmetry, vertex, domain and range. The steps to graph a quadratic function are given as finding the axis of symmetry, the vertex, and then two other points to reflect across the axis and connect with a smooth curve. Methods for finding the axis of symmetry, vertex, y-intercept and solutions are demonstrated through examples.

rational equation transformable to quadratic equation.pptx

1. The document provides examples for solving quadratic equations that are not in standard form by transforming them into standard form ax2 + bx + c = 0 and then using methods like factoring or the quadratic formula.
2. It also gives examples for solving rational algebraic equations by multiplying both sides by the least common denominator to obtain a quadratic equation, transforming it into standard form, and then solving.
3. The examples cover topics like solving for the solution set, checking solutions, and using the quadratic formula to solve transformed equations.

Solving Quadratic Equations

Provides example on how to solve quadratic equations using extracting square roots, factoring, completing the square and quadratic formula.

55 addition and subtraction of rational expressions

The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.

Polynomial equations

This document discusses finding the real roots of polynomial equations. It states that a polynomial of degree n can have at most n real roots. It then provides examples of factorizing polynomials into their linear factors to find their roots. Finally, it lists 5 polynomial equations and asks the reader to find the roots of each by factorizing.

Synthetic division

This document provides instructions for using synthetic division to divide polynomials. It contains the following key points:
1. Synthetic division can be used to divide polynomials when the divisor has a leading coefficient of 1 and there is a coefficient for every power of the variable in the numerator.
2. The procedure involves writing the terms of the numerator in descending order, bringing down the constant of the divisor, multiplying and adding down the columns to obtain the coefficients of the quotient polynomial and the remainder.
3. An example problem walks through each step of synthetic division to divide (5x^4 - 4x^2 + x + 6) / (x - 3), obtaining a quotient of 5x^3 + 15

7.7 Solving Radical Equations

This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.

Zeros of a polynomial function

The document discusses polynomial functions and their roots. It begins by defining that the roots of a polynomial function are the values of x that make the function equal to 0. It then provides examples of finding the roots of linear and quadratic equations. Next, it introduces the Rational Root Theorem, which states that possible rational roots must be factors of the constant term and leading coefficient. Examples are given to demonstrate applying the theorem. The document concludes by using synthetic division to find all three roots of a cubic polynomial given one known root.

Graphing Quadratics

This document provides information about graphing quadratic functions. It defines the standard form of a quadratic function as y = ax^2 + bx + c and explains that the sign of a determines if the parabola opens up or down. It describes how to find the axis of symmetry, vertex, domain and range. The steps to graph a quadratic function are given as finding the axis of symmetry, the vertex, and then two other points to reflect across the axis and connect with a smooth curve. Methods for finding the axis of symmetry, vertex, y-intercept and solutions are demonstrated through examples.

Mathematics 9 Lesson 1-B: Solving Quadratic Equations using Quadratic Formula

This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using the Quadratic Formula. It also discusses the steps in solving quadratic equations using the method of Quadratic Formula.

The remainder theorem powerpoint

The document explains the Remainder Theorem in multiple ways using different examples and proofs. It states that the Remainder Theorem provides a test to determine if a polynomial f(x) is divisible by a polynomial of the form x-c. It proves that the remainder obtained when dividing f(x) by x-c is equal to the value of f(x) when x is substituted with c. It provides multiple examples working through applying the Remainder Theorem to determine if various polynomials are divisible.

Nature of the roots and sum and product of the roots of a quadratic equation

Provides example and solutions about about how to find the roots of a quadratic equation and its sum and product.

Factoring Sum and Difference of Two Cubes

This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u

Lesson 1: Special Products

This document provides examples and rules for working with exponents and polynomials. It begins by showing the step-by-step working of three multiplication problems involving exponents. It then states the product rule for exponents and the rule for raising a power to another power. The document encourages positive thinking and hard work. It ends by having the reader say a phrase aloud together to reinforce a growth mindset towards math.

Polynomial function

The document defines a polynomial function as a function of the form f(x) = anxn + an-1xn-1 +...+ a0, where n is a nonnegative integer and an, an-1,...a0 are real numbers with an ≠ 0. The degree of a polynomial is the highest exponent of its terms. Examples are provided to illustrate how to determine the degree and number of terms of polynomial functions. The document also asks questions to check understanding of identifying polynomial functions and determining their degree.

Modeling with Quadratics

- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.

factoring polynomials

This document outlines key concepts and examples for factoring polynomials. It covers factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL multiplication. Sections include the greatest common factor, factoring trinomials of the forms x^2 + bx + c and ax^2 + bx + c.

Graphs of polynomial functions

The document summarizes key characteristics of polynomial functions:
1) Polynomial functions produce smooth, continuous curves on their domains which are the set of real numbers.
2) The graph's x-intercepts, turning points, and absolute/relative maxima and minima are defined.
3) As the degree of a polynomial increases, so do the possible number of x-intercepts and turning points, up to the degree value. The leading coefficient and degree determine whether the graph rises or falls.

Polynomial equations

Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.

Linear Equations in Two Variables

The document discusses linear equations in two variables. It defines a linear equation as one that can be written in the standard form Ax + By = C, where A, B, and C are real numbers and A and B cannot both be zero. Examples are provided of determining if equations are linear and identifying the A, B, and C components if they are linear. The document also discusses using ordered pairs as solutions to linear equations and finding multiple solutions to a given linear equation.

Inverse function

The document discusses inverse functions, including:
- An inverse function undoes the output of the original function by relating the input and output variables.
- For a function to have an inverse, it must be one-to-one so that each output is paired with a unique input.
- To find the inverse of a function, swap the input and output variables and isolate the new output variable.

Dividing Polynomials Slide Share

I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...

nature of the roots and discriminant

This document provides guidance on identifying the nature of roots of quadratic equations. It begins by identifying the least mastered skill of identifying the nature of roots. It then reviews the standard form of a quadratic equation and the quadratic formula. The key concept of the discriminant is explained, which determines the number and type of roots. Examples are provided to show how to find the discriminant and use it to describe the nature of roots. Activities are included for students to practice rewriting equations in standard form, finding a, b, c values, calculating discriminants, and determining the nature of roots. An assessment card with example problems is provided to check understanding.

Lesson 10: The Chain Rule

The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.

Factoring Quadratic Trinomials

The document provides instructions for factoring quadratic trinomials using 4 examples. It explains that you write the trinomial as two parentheses, factor the constant term into the parentheses, then check that the factors give the middle term of the original expression when multiplied out. The process involves 4 steps and is demonstrated factoring expressions like x^2 + 10x + 24 and x^2 - 8x + 15.

Factoring with Common Monomial Factor

You will learn how to factor polynomials with common monomial factor.
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u

Quadratic Equation and discriminant

This document introduces the quadratic formula as a method for solving quadratic equations. It shows the steps for deriving the formula from completing the square and provides examples of its use. The discriminant is defined as b^2 - 4ac from the quadratic formula. The sign of the discriminant determines the number and type of roots: positive discriminant yields two real roots, zero discriminant yields one real root, and negative discriminant yields two complex roots. Examples are provided to illustrate each case.

Mathematics 9 Lesson 3: Quadratic Functions

This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.

Rational Root Theorem

The Rational Root Theorem provides a method to find all possible rational roots of a polynomial with integer coefficients. It states that every rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Examples are provided to demonstrate finding all possible rational roots using this theorem and then checking them to determine the actual rational roots. Once a rational root is found, synthetic division can be used to find the depressed polynomial which can then be fully factored to obtain all factors of the original polynomial.

Expresiones algebraicas

The document discusses various algebraic expressions and operations on them including:
- Summing and subtracting monomials and polynomials
- Multiplying monomials, polynomials, and binomial expressions using rules like distributing terms
- Dividing monomials and polynomials by breaking them into fractions
- Evaluating numerical values of expressions by substituting values for variables
- Special multiplication rules for perfect square binomials, difference of squares, and conjugates
- Multiplying polynomials with a common term using a specific formula

Factoring polynomials

Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.

Mathematics 9 Lesson 1-B: Solving Quadratic Equations using Quadratic Formula

This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using the Quadratic Formula. It also discusses the steps in solving quadratic equations using the method of Quadratic Formula.

The remainder theorem powerpoint

The document explains the Remainder Theorem in multiple ways using different examples and proofs. It states that the Remainder Theorem provides a test to determine if a polynomial f(x) is divisible by a polynomial of the form x-c. It proves that the remainder obtained when dividing f(x) by x-c is equal to the value of f(x) when x is substituted with c. It provides multiple examples working through applying the Remainder Theorem to determine if various polynomials are divisible.

Nature of the roots and sum and product of the roots of a quadratic equation

Provides example and solutions about about how to find the roots of a quadratic equation and its sum and product.

Factoring Sum and Difference of Two Cubes

This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u

Lesson 1: Special Products

This document provides examples and rules for working with exponents and polynomials. It begins by showing the step-by-step working of three multiplication problems involving exponents. It then states the product rule for exponents and the rule for raising a power to another power. The document encourages positive thinking and hard work. It ends by having the reader say a phrase aloud together to reinforce a growth mindset towards math.

Polynomial function

The document defines a polynomial function as a function of the form f(x) = anxn + an-1xn-1 +...+ a0, where n is a nonnegative integer and an, an-1,...a0 are real numbers with an ≠ 0. The degree of a polynomial is the highest exponent of its terms. Examples are provided to illustrate how to determine the degree and number of terms of polynomial functions. The document also asks questions to check understanding of identifying polynomial functions and determining their degree.

Modeling with Quadratics

- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.

factoring polynomials

This document outlines key concepts and examples for factoring polynomials. It covers factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL multiplication. Sections include the greatest common factor, factoring trinomials of the forms x^2 + bx + c and ax^2 + bx + c.

Graphs of polynomial functions

The document summarizes key characteristics of polynomial functions:
1) Polynomial functions produce smooth, continuous curves on their domains which are the set of real numbers.
2) The graph's x-intercepts, turning points, and absolute/relative maxima and minima are defined.
3) As the degree of a polynomial increases, so do the possible number of x-intercepts and turning points, up to the degree value. The leading coefficient and degree determine whether the graph rises or falls.

Polynomial equations

Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.

Linear Equations in Two Variables

The document discusses linear equations in two variables. It defines a linear equation as one that can be written in the standard form Ax + By = C, where A, B, and C are real numbers and A and B cannot both be zero. Examples are provided of determining if equations are linear and identifying the A, B, and C components if they are linear. The document also discusses using ordered pairs as solutions to linear equations and finding multiple solutions to a given linear equation.

Inverse function

The document discusses inverse functions, including:
- An inverse function undoes the output of the original function by relating the input and output variables.
- For a function to have an inverse, it must be one-to-one so that each output is paired with a unique input.
- To find the inverse of a function, swap the input and output variables and isolate the new output variable.

Dividing Polynomials Slide Share

I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...

nature of the roots and discriminant

This document provides guidance on identifying the nature of roots of quadratic equations. It begins by identifying the least mastered skill of identifying the nature of roots. It then reviews the standard form of a quadratic equation and the quadratic formula. The key concept of the discriminant is explained, which determines the number and type of roots. Examples are provided to show how to find the discriminant and use it to describe the nature of roots. Activities are included for students to practice rewriting equations in standard form, finding a, b, c values, calculating discriminants, and determining the nature of roots. An assessment card with example problems is provided to check understanding.

Lesson 10: The Chain Rule

The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.

Factoring Quadratic Trinomials

The document provides instructions for factoring quadratic trinomials using 4 examples. It explains that you write the trinomial as two parentheses, factor the constant term into the parentheses, then check that the factors give the middle term of the original expression when multiplied out. The process involves 4 steps and is demonstrated factoring expressions like x^2 + 10x + 24 and x^2 - 8x + 15.

Factoring with Common Monomial Factor

You will learn how to factor polynomials with common monomial factor.
For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u

Quadratic Equation and discriminant

This document introduces the quadratic formula as a method for solving quadratic equations. It shows the steps for deriving the formula from completing the square and provides examples of its use. The discriminant is defined as b^2 - 4ac from the quadratic formula. The sign of the discriminant determines the number and type of roots: positive discriminant yields two real roots, zero discriminant yields one real root, and negative discriminant yields two complex roots. Examples are provided to illustrate each case.

Mathematics 9 Lesson 3: Quadratic Functions

This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.

Rational Root Theorem

The Rational Root Theorem provides a method to find all possible rational roots of a polynomial with integer coefficients. It states that every rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Examples are provided to demonstrate finding all possible rational roots using this theorem and then checking them to determine the actual rational roots. Once a rational root is found, synthetic division can be used to find the depressed polynomial which can then be fully factored to obtain all factors of the original polynomial.

Mathematics 9 Lesson 1-B: Solving Quadratic Equations using Quadratic Formula

Mathematics 9 Lesson 1-B: Solving Quadratic Equations using Quadratic Formula

The remainder theorem powerpoint

The remainder theorem powerpoint

Nature of the roots and sum and product of the roots of a quadratic equation

Nature of the roots and sum and product of the roots of a quadratic equation

Factoring Sum and Difference of Two Cubes

Factoring Sum and Difference of Two Cubes

Lesson 1: Special Products

Lesson 1: Special Products

Polynomial function

Polynomial function

Modeling with Quadratics

Modeling with Quadratics

factoring polynomials

factoring polynomials

Graphs of polynomial functions

Graphs of polynomial functions

Polynomial equations

Polynomial equations

Linear Equations in Two Variables

Linear Equations in Two Variables

Inverse function

Inverse function

Dividing Polynomials Slide Share

Dividing Polynomials Slide Share

nature of the roots and discriminant

nature of the roots and discriminant

Lesson 10: The Chain Rule

Lesson 10: The Chain Rule

Factoring Quadratic Trinomials

Factoring Quadratic Trinomials

Factoring with Common Monomial Factor

Factoring with Common Monomial Factor

Quadratic Equation and discriminant

Quadratic Equation and discriminant

Mathematics 9 Lesson 3: Quadratic Functions

Mathematics 9 Lesson 3: Quadratic Functions

Rational Root Theorem

Rational Root Theorem

Expresiones algebraicas

The document discusses various algebraic expressions and operations on them including:
- Summing and subtracting monomials and polynomials
- Multiplying monomials, polynomials, and binomial expressions using rules like distributing terms
- Dividing monomials and polynomials by breaking them into fractions
- Evaluating numerical values of expressions by substituting values for variables
- Special multiplication rules for perfect square binomials, difference of squares, and conjugates
- Multiplying polynomials with a common term using a specific formula

Factoring polynomials

Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.

Algebra slideshow

The document provides instructions for graphing and solving various types of quadratic equations. It defines standard form, vertex form, and intercept form of quadratics. It explains how to graph quadratics by finding the vertex and intercepts. Methods covered include factoring, taking square roots, completing the square, and using the quadratic formula. Examples are included to demonstrate each process.

P6 factoring

This document discusses various techniques for factoring algebraic expressions, including:
1) Factoring out common factors from terms.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose sum is b and product is c.
3) Using special formulas to factor differences and sums of squares and cubes.
4) Recognizing perfect squares and factoring them.
5) Factoring completely by repeated application of factoring methods.

P6 factoring

This document discusses various techniques for factoring algebraic expressions, including:
1) Factoring out common factors from terms.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose sum is b and product is c.
3) Using special formulas to factor differences and sums of squares and cubes.
4) Recognizing perfect squares and factoring them.
5) Factoring completely by repeated application of factoring methods.

Paso 2 profundizar y contextualizar el conocimiento de la unidad 1 ff

This document provides an overview of algebraic expressions and factorization techniques covered in Unit 1. It defines basic algebraic expressions, polynomials, monomials, binomials, trinomials and more. It then describes 10 cases of factorization, including common factors, difference of squares, sum/difference of cubes, and more. Examples are provided for each case to demonstrate the factorization processes.

Factorisation 140814105901-phpapp02

This document outlines the key topics and methods for factorizing algebraic expressions covered in a maths project. It includes factorizing natural numbers and algebraic expressions, using the method of common factors, factorizing by regrouping terms, solving sums using factorizing expressions and identities, factors of the form (x+a)(x+b), and dividing algebraic expressions including monomials and polynomials. Examples are provided for each method to demonstrate how to factorize different algebraic expressions step-by-step.

Mock cat solutions paper no 1

This document contains a mock CAT exam with multiple choice questions and explanations. It consists of two pages. The first page lists 60 multiple choice questions with answer options A-D. The second page provides explanations for the questions and solutions to problems. It discusses topics like probability, ratios, geometry, time/speed/distance word problems, and data interpretation from graphs.

Algebra unit 8.7

This document provides examples and explanations for factoring special cases of polynomials, including perfect square trinomials and the difference of two squares. It includes examples of recognizing, factoring, and explaining if expressions are in one of these forms. One example problem finds the perimeter of a garden given its area as a factored expression and evaluates the perimeter for a given value of x.

March 10

The document provides examples of factoring quadratic expressions using different techniques like finding the greatest common factor, difference of squares, and perfect square trinomials. It also reviews the formulas for factoring the square of a sum and square of a difference and provides a test review with additional practice problems. The review is intended to prepare students to factor a variety of quadratic expressions using various factoring methods.

3.2 factoring polynomials

This document provides instruction on factoring polynomials. It begins with examples of factoring linear and quadratic expressions. It then discusses using the Factor Theorem and the Remainder Theorem to determine if a binomial is a factor of a polynomial. Additional examples demonstrate factoring polynomials by grouping like terms and using special rules to factor the sum and difference of cubes. An example applies these factoring techniques to model the volume of a storage box.

Chapter 2

This document provides an overview of quadratic equations and inequalities. It defines quadratic equations as equations of the form ax2 + bx + c = 0, where a, b, and c are real number constants and a ≠ 0. Examples of quadratic equations are provided. Methods for solving quadratic equations are discussed, including factoring, completing the square, and the quadratic formula. Properties of inequalities are outlined. The chapter also covers solving polynomial and rational inequalities, as well as equations and inequalities involving absolute value. Practice problems are included at the end.

Matematica

The document provides definitions and explanations of key concepts in algebra including:
- Sum, subtraction, multiplication, division and their properties
- Evaluating algebraic expressions by substituting numeric values
- Factorization using techniques like difference of squares, difference of cubes, and perfect squares
- Products notable including formulas for (a+b)2, (a-b)2, (a+b)3 and (a-b)3
- Examples of simplifying algebraic expressions using definitions and properties like distributive, commutative, associative properties.

Factoring

The document discusses various factoring techniques, including:
- Factoring by greatest common factor (GCF)
- Factoring trinomials of the form ax^2 + bx + c by finding two numbers that multiply to c and add to b
- Factoring the difference of two squares and perfect square trinomials
- Using the "box method" to factor trinomials by placing factors of the first and last terms in the boxes
- Factoring by grouping polynomials with four terms into two groups of two terms each.

Module 3 quadratic functions

This module covers quadratic functions and equations. Students will learn to determine the zeros of quadratic functions by relating them to the roots of quadratic equations. They will also learn to find the roots of quadratic equations using factoring, completing the square, and the quadratic formula. The module aims to help students derive quadratic functions given certain conditions like the zeros, a table of values, or a graph.

GR 8 Math Powerpoint about Polynomial Techniques

-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.

Quadratic equations

This document discusses quadratic equations and methods for solving them. It begins by defining quadratic equations as second degree polynomial equations of the form ax^2 + bx + c = 0, where a is not equal to 0. It then presents several methods for finding the roots or solutions of quadratic equations: factoring, completing the square, and using the quadratic formula. Examples are provided to illustrate each method. The document also discusses graphing quadratic functions and key features of parabolas such as vertex, axis of symmetry, and direction of opening.

Perfect square of Binomials

1. The square of a binomial (a + b) is a trinomial with terms a2, 2ab, and b2.
2. To square a binomial, square each term and multiply the unlike terms by 2.
3. Examples are provided of squaring binomials like (x + 6)2 = x2 + 12x + 36 and factoring trinomials into perfect square forms like (x - 2)2.

Chithra

This document provides an overview of different methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. It includes examples of solving quadratic equations using each method. Factorisation involves finding two binomial factors whose product is the quadratic expression. Completing the square transforms the equation into a perfect square plus an extra term, allowing it to be factorised. The quadratic formula provides the general solution for any quadratic equation in the form ax^2 + bx + c = 0.

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Algebra unit 8.7

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GR 8 Math Powerpoint about Polynomial Techniques

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The document discusses using the two intercept form to find the equation of a line given two points. It provides the two intercept form equation, where a and b are the x and y intercepts. It then works through three examples of finding the line equation using two points and the two intercept form. It lists additional practice problems and their solutions for finding line equations using two points and the two intercept form.

equation of the line using two point form

This document discusses using the two-point form to find the equation of a line given two points. It provides the two-point form equation, examples of using the form to find the slope and y-intercept of lines, and practice problems for determining the equation of lines passing through two points. The goal is to determine the equation in slope-intercept form using the two-point form equation and substituting the x- and y-coordinates of the two points.

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This document discusses using the point-slope form to find the equation of a line given a slope and point. It provides the point-slope form equation, and examples of finding the line equation for different slopes and points. Exercises are provided for the reader to practice finding additional line equations using given slopes and points.

writing linear equation

This document provides examples for rewriting linear equations between the slope-intercept form (y=mx+b) and standard form (Ax + By = C).
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common monomial factor

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elimination

elimination

factoring perfect square trinomial

factoring perfect square trinomial

two intercept form

two intercept form

equation of the line using two point form

equation of the line using two point form

point slope form

point slope form

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common monomial factor

Revised guidelines on the use of the Special Education Fund

Revised guidelines on the use of the Special Education Fund

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- 1. FACTORS OF POLYNOMIALS Factoring the Difference of Two Squares
- 2. Jessebel G. Bautista Antonio J. Villegas Voc’l High School My Profile
- 3. OBJECTIVES • Identify the perfect squares • Factor the difference of two squares • Factor with accuracy
- 4. Factoring a polynomial involves writing it as a product of two or more polynomials. It reverses the process of polynomial multiplication. Every polynomial that is a difference of squares can be factored by applying the following formula: a2 – b2 = (a + b) (a – b)
- 5. Every polynomial that is a difference of squares can be factored by applying the following formula: a2 – b2 = (a + b) (a – b) Note that a and b in the pattern can be any algebraic expression. For example, for a = x , and b=2 we get the following: x2 – 22 = (x – 2) (x + 2) The polynomial x2 – 4 is now expressed in factored form, (x+2)(x−2). We can expand the right-hand side of this equation to justify the factorization: (x + 2) (x – 2) = x(x – 2) + 2(x – 2) = x2 – 2x + 2x – 4 = x2 – 4 Now that we understand the pattern, let's use it to factor a few more polynomials.
- 6. Both x2 and 16 are perfect square since x2 = (x)2 and 16 = (4)2. In other words. x2 – 16 = (x)2 – (4)2 EXAMPLE 1: Factoring x2 - 16 Since the two squares are being subtracted, we can see that this polynomial represents a difference of squares. We can use the difference of squares pattern to factor this expression: a2 – b2 = (a + b) (a – b) In our case, a = x and b = 4. Therefore, our polynomial factors as follows: (x)2 – (4)2 = (x – 4) (x + 4) We can check our work by ensuring the product of these two factors is x2 – 16 .
- 7. Both 9x2 and 100 are perfect square since 9x2 = (3x)2 and 100 = (10)2. In other words. 9x2 – 100 = (3x)2 – (10)2 EXAMPLE 2: Factoring 9x2 - 100 Since the two squares are being subtracted, we can see that this polynomial represents a difference of squares. We can use the difference of squares pattern to factor this expression: a2 – b2 = (a + b) (a – b) In our case, a = 3x and b = 10. Therefore, our polynomial factors as follows: (3x)2 – (10)2 = (3x – 10) (3x + 10) We can check our work by ensuring the product of these two factors is 9x2 – 100 .
- 8. Both 36m2 and 49n4 are perfect square since 36m2 = (6m)2 and 49n4 = (7n2)2. In other words. 36m2 – 49n4 = (6m)2 – (7n2)2 EXAMPLE 3: Factoring 36m2 – 49n4 Since the two squares are being subtracted, we can see that this polynomial represents a difference of squares. We can use the difference of squares pattern to factor this expression: a2 – b2 = (a + b) (a – b) In our case, a = 6m and b = 7n2. Therefore, our polynomial factors as follows: (6m)2 – (7n2)2 = (6m – 7n2) (6m + 7n2) We can check our work by ensuring the product of these two factors is 36m2 – 49n4 .
- 9. 1. 4x2 – 25 2. 9y2 – 100 3. 49m2 – 121n2 4. 144 – 16n2 5. m2n2 – 196 TO DO … Factor the following: 6. 169a2 – b2c2 7. 9x2 – 81 8. 289m2 – 256n2 9. 16b2 – 9c2 10. 121c4 – 169d2