Copyright © Cengage Learning. All rights reserved. Factoring Polynomials and Solving Equations by Factoring 5

Copyright © Cengage Learning. All rights reserved. Section 5.5 Factoring the Sum and Difference of Two Cubes

3 Objectives Factor the sum of two cubes. Factor the difference of two cubes. Factor a polynomial involving the sum or difference of two cubes

4 Factoring the Sum and Difference of Two Cubes Recall that the difference of the squares of two quantities factors into the product of two binomials. One binomial is the sum of the quantities, and the other is the difference of the quantities. x 2 – y 2 = (x + y)(x – y) or F 2 – L 2 = (F + L)(F – L) In this section, we will discuss formulas for factoring the sum of two cubes and the difference of two cubes.

5 Factor the sum of two cubes 1.

6 Factor the sum of two cubes To discover the pattern for factoring the sum of two cubes, we find the following product: (x + y)(x 2 – xy + y 2 ) = x 3 – x 2 y + xy 2 + x 2 y– xy 2 + y 3 = x 3 + y 3 This result justifies the formula for factoring the sum of two cubes. Use the distributive property. Combine like terms.

7 Factoring the Sum of Two Cubes x 3 + y 3 = (x + y)(x 2 – xy + y 2 ) To factor the sum of two cubes, it is helpful to know the cubes of the numbers from 1 to 10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000 Expressions containing variables such as x 6 y 3 are also perfect cubes, because they can be written as the cube of a quantity: x 6 y 3 = (x 2 y) 3 Factor the sum of two cubes

8 Factor: x Solution: The binomial x is the sum of two cubes, because x = x Thus, x factors as (x + 2) times the trinomial (x 2 – 2  x ). x = x = (x + 2)(x 2 – x  ) Example

9 = (x + 2)(x 2 – 2x + 4) To check, we can use the distributive property and combine like terms. (x + 2)(x 2 – 2x + 4) = x 3 – 2x 2 + 4x + 2x 2 – 4x + 8 = x 3 – 8 cont’d Example – Solution

10 Factor the difference of two cubes 2.

11 Factor the difference of two cubes To discover the pattern for factoring the difference of two cubes, we find the following product: (x – y)(x 2 + xy + y 2 ) = x 3 + x 2 y + xy 2 – x 2 y – xy 2 – y 3 = x 3 – y 3 This result justifies the formula for factoring the difference of two cubes. Use the distributive property. Combine like terms.

12 Factor the difference of two cubes Factoring the Difference of Two Cubes x 3 – y 3 = (x – y)(x 2 + xy + y 2 )

13 Factor: a 3 – 64b 3. Solution: The binomial a 3 – 64b 3 is the difference of two cubes. a 3 – 64b 3 = a 3 – (4b) 3 Thus, its factors are the difference a – 4b and the trinomial a 2 + a(4b) + (4b) 2. a 3 – 64b 3 = a 3 – (4b) 3 = (a – 4b)[a 2 + a(4b) + (4b) 2 ] Example

14 = (a – 4b)(a 2 + 4ab + 16b 2 ) To check, we can use the distributive property and combine like terms. (a – 4b)(a 2 + 4ab + 16b 2 ) = a 3 + 4a 2 b + 16ab 2 – 4a 2 b – 16ab 2 – 64b 3 = a 3 – 64b 3 cont’d Example – Solution

15 Factor a polynomial involving the sum or difference of two cubes 3.

16 Factor a polynomial involving the sum or difference of two cubes Sometimes we must factor out a greatest common factor before factoring a sum or difference of two cubes.

17 Example Factor: –2t t 2. Solution: –2t t 2 = –2t 2 (t 3 – 64) = –2t 2 (t 3 – 4 3 ) = –2t 2 (t – 4)(t 2 + 4t + 16) We can check by multiplication. Factor the GCF, –2t 2, from (–2t t 2 ). Factor t 3 – 4 3. Write t 3 – 64 as t 3 – 4 3.