Learn on PengiBig Ideas Math, Algebra 1Chapter 7: Polynomial Equations and Factoring

Lesson 7: Factoring Special Products

Property 1. $a^2 + 2ab + b^2 = (a + b)^2$.

Section 1

Factoring Special Products

Property

$a^2 + 2ab + b^2 = (a + b)^2$

$a^2 - 2ab + b^2 = (a - b)^2$

$a^2 - b^2 = (a + b)(a - b)$

Section 2

Factor Differences of Squares

Property

If $a$ and $b$ are real numbers, a difference of squares factors to a product of conjugates using the following pattern:

a^2 - b^2 = (a-b)(a+b)

To use this pattern, ensure you have a binomial where two perfect squares are being subtracted.
Write each term as a square,

(a)^2 - (b)^2

, then write the product of the conjugates,

(a-b)(a+b)

.
Note that a sum of squares,

a^2+b^2

, is prime and cannot be factored.

Examples

To factor $9x^2 - 25$ , rewrite the expression as a difference of squares, $(3x)^2 - 5^2$ . This factors into the product of conjugates $(3x-5)(3x+5)$ .

To factor $16a^2 - 81b^2$ , identify the terms as $(4a)^2$ and $(9b)^2$ . The factored form is the product of their conjugates, $(4a-9b)(4a+9b)$ .

Book overview

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Chapter 7: Polynomial Equations and Factoring

Back to Big Ideas Math, Algebra 1

Lesson 1
Lesson 1: Adding and Subtracting Polynomials
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Lesson 2
Lesson 2: Multiplying Polynomials
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Lesson 3
Lesson 3: Special Products of Polynomials
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Lesson 4
Lesson 4: Solving Polynomial Equations in Factored Form
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Lesson 5
Lesson 5: Factoring x² + bx + c
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Lesson 6
Lesson 6: Factoring ax² + bx + c
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Lesson 7Current
Lesson 7: Factoring Special Products
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Lesson 8
Lesson 8: Factoring Polynomials Completely
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Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Factoring Special Products

Property

$a^2 + 2ab + b^2 = (a + b)^2$

$a^2 - 2ab + b^2 = (a - b)^2$

$a^2 - b^2 = (a + b)(a - b)$

Section 2

Factor Differences of Squares

Property

If $a$ and $b$ are real numbers, a difference of squares factors to a product of conjugates using the following pattern:

a^2 - b^2 = (a-b)(a+b)

To use this pattern, ensure you have a binomial where two perfect squares are being subtracted.
Write each term as a square,

(a)^2 - (b)^2

, then write the product of the conjugates,

(a-b)(a+b)

.
Note that a sum of squares,

a^2+b^2

, is prime and cannot be factored.

Examples

To factor $9x^2 - 25$ , rewrite the expression as a difference of squares, $(3x)^2 - 5^2$ . This factors into the product of conjugates $(3x-5)(3x+5)$ .

To factor $16a^2 - 81b^2$ , identify the terms as $(4a)^2$ and $(9b)^2$ . The factored form is the product of their conjugates, $(4a-9b)(4a+9b)$ .

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 7: Polynomial Equations and Factoring

Back to Big Ideas Math, Algebra 1

Lesson 1
Lesson 1: Adding and Subtracting Polynomials
Learn Practice
Lesson 2
Lesson 2: Multiplying Polynomials
Learn Practice
Lesson 3
Lesson 3: Special Products of Polynomials
Learn Practice
Lesson 4
Lesson 4: Solving Polynomial Equations in Factored Form
Learn Practice
Lesson 5
Lesson 5: Factoring x² + bx + c
Learn Practice
Lesson 6
Lesson 6: Factoring ax² + bx + c
Learn Practice
Lesson 7Current
Lesson 7: Factoring Special Products
Learn Practice
Lesson 8
Lesson 8: Factoring Polynomials Completely
Learn Practice

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Factoring Special Products

Property

$a^2 + 2ab + b^2 = (a + b)^2$

$a^2 - 2ab + b^2 = (a - b)^2$

$a^2 - b^2 = (a + b)(a - b)$

Section 2

Factor Differences of Squares

Property

If $a$ and $b$ are real numbers, a difference of squares factors to a product of conjugates using the following pattern:

a^2 - b^2 = (a-b)(a+b)

To use this pattern, ensure you have a binomial where two perfect squares are being subtracted.
Write each term as a square,

(a)^2 - (b)^2

, then write the product of the conjugates,

(a-b)(a+b)

.
Note that a sum of squares,

a^2+b^2

, is prime and cannot be factored.

Examples

To factor $9x^2 - 25$ , rewrite the expression as a difference of squares, $(3x)^2 - 5^2$ . This factors into the product of conjugates $(3x-5)(3x+5)$ .

To factor $16a^2 - 81b^2$ , identify the terms as $(4a)^2$ and $(9b)^2$ . The factored form is the product of their conjugates, $(4a-9b)(4a+9b)$ .

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 7: Polynomial Equations and Factoring

Back to Big Ideas Math, Algebra 1

Lesson 1
Lesson 1: Adding and Subtracting Polynomials
Learn Practice
Lesson 2
Lesson 2: Multiplying Polynomials
Learn Practice
Lesson 3
Lesson 3: Special Products of Polynomials
Learn Practice
Lesson 4
Lesson 4: Solving Polynomial Equations in Factored Form
Learn Practice
Lesson 5
Lesson 5: Factoring x² + bx + c
Learn Practice
Lesson 6
Lesson 6: Factoring ax² + bx + c
Learn Practice
Lesson 7Current
Lesson 7: Factoring Special Products
Learn Practice
Lesson 8
Lesson 8: Factoring Polynomials Completely
Learn Practice

Lesson 7: Factoring Special Products

Property

Property

Examples

Chapter 7: Polynomial Equations and Factoring

Lesson 1: Adding and Subtracting Polynomials

Lesson 2: Multiplying Polynomials

Lesson 3: Special Products of Polynomials

Lesson 4: Solving Polynomial Equations in Factored Form

Lesson 5: Factoring x² + bx + c

Lesson 6: Factoring ax² + bx + c

Lesson 7: Factoring Special Products

Lesson 8: Factoring Polynomials Completely

Lesson overview

Property

Property

Examples

Chapter 7: Polynomial Equations and Factoring

Lesson 1: Adding and Subtracting Polynomials

Lesson 2: Multiplying Polynomials

Lesson 3: Special Products of Polynomials

Lesson 4: Solving Polynomial Equations in Factored Form

Lesson 5: Factoring x² + bx + c

Lesson 6: Factoring ax² + bx + c

Lesson 7: Factoring Special Products

Lesson 8: Factoring Polynomials Completely

Lesson 1: Adding and Subtracting Polynomials

Lesson 2: Multiplying Polynomials

Lesson 3: Special Products of Polynomials

Lesson 4: Solving Polynomial Equations in Factored Form

Lesson 5: Factoring x² + bx + c

Lesson 6: Factoring ax² + bx + c

Lesson 7: Factoring Special Products

Lesson 8: Factoring Polynomials Completely