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

  1. a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2
  1. a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2
  1. a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

Section 2

Factor Differences of Squares

Property

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

a2−b2=(a−b)(a+b)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(a)^2 - (b)^2, then write the product of the conjugates, (a−b)(a+b)(a-b)(a+b).
Note that a sum of squares, a2+b2a^2+b^2, is prime and cannot be factored.

Examples

  • To factor 9x2−259x^2 - 25, rewrite the expression as a difference of squares, (3x)2−52(3x)^2 - 5^2. This factors into the product of conjugates (3x−5)(3x+5)(3x-5)(3x+5).
  • To factor 16a2−81b216a^2 - 81b^2, identify the terms as (4a)2(4a)^2 and (9b)2(9b)^2. The factored form is the product of their conjugates, (4a−9b)(4a+9b)(4a-9b)(4a+9b).

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Factoring Special Products

Property

  1. a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2
  1. a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2
  1. a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

Section 2

Factor Differences of Squares

Property

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

a2−b2=(a−b)(a+b)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(a)^2 - (b)^2, then write the product of the conjugates, (a−b)(a+b)(a-b)(a+b).
Note that a sum of squares, a2+b2a^2+b^2, is prime and cannot be factored.

Examples

  • To factor 9x2−259x^2 - 25, rewrite the expression as a difference of squares, (3x)2−52(3x)^2 - 5^2. This factors into the product of conjugates (3x−5)(3x+5)(3x-5)(3x+5).
  • To factor 16a2−81b216a^2 - 81b^2, identify the terms as (4a)2(4a)^2 and (9b)2(9b)^2. The factored form is the product of their conjugates, (4a−9b)(4a+9b)(4a-9b)(4a+9b).