The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

Name Date

WORKSHEET 5.19: FACTORING THE SUMS AND DIFFERENCES

OF CUBES

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Binomials that are the sums and differences of cubes can be factored by using these

formulas:

Sum of cubes Difference of cubes

a^3 +b^3 =(a+b)(a^2 −ab+b^2 ) a^3 −b^3 =(a−b)(a^2 +ab+b^2 )

To use these formulas, follow the steps below:

1. Identify the cubed terms in each binomial.


2. Substitute the appropriate variable foraandbin the correct formula.


3. Check your work by multiplying the factors.

EXAMPLE

Factorx^3 − 27.

xis cubed and 27 is equal to 33. The expression is the difference of cubes.

Use the formula for finding the difference of cubes. Substitutexforaand 3 forbin the

formula.x^3 − 33 =(x−3)(x^2 + 3 x+9)

Check:x(x^2 + 3 x+9)−3(x^2 + 3 x+9)=x^3 + 3 x^2 + 9 x− 3 x^2 − 9 x− 27 =x^3 − 27

DIRECTIONS: Factor each binomial. If the binomial cannot be factored, write ‘‘cannot be

factored.’’

1. x^3 − 8 2. x^3 + 125 3. x^3 + 32


2. x^3 − 1 5. 64 −y^3 6. x^3 y^3 + 512

CHALLENGE:Jasmine factoredx^3 +27 as (x+3)(x^2 − 3 x+9). She used FOIL

to check the factors and foundx^3 + 9 x+ 3 x^2 +27. Her work did not check.

What was her mistake?

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2011 by Judith A. Muschla, Gary Robert Muschla, and Erin Muschla. All rights reserved.