1. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 5-1 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 5 Polynomials and Factoring

2. 5-2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Factoring Sums or Differences of Cubes 5.5

3. 5-3 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Factoring Sums or Differences of Cubes We have seen that a difference of two squares can always be factored, but a sum of two squares is usually prime. The difference or sum of two cubes can always be factored

4. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 5-4 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Factoring a Sum or Difference of Two Cubes A 3 + B 3 = (A + B)(A 2 – AB + B 2 ); A 3 – B 3 = (A – B)(A 2 + AB + B 2 )

5. 5-5 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Solution x 3 – 8. Write an equivalent expression by factoring: A 3 – B 3 = (A – B)(A 2 + AB + B 2 ) First observe that x 3 – 8 = x 3 – 2 3 From the formula: We have x 3 – 2 3 = (x – 2)(x 2 + 2x + 2 2 ) = (x – 2)(x 2 + 2x + 4)

6. 5-6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Solution b 6 + 125y 3. Write an equivalent expression by factoring: A 3 + B 3 = (A + B)(A 2 – AB + B 2 ) First observe that b 6 + 125y 3 = (b 2 ) 3 + (5y) 3 From the formula: We have (b 2 ) 3 + (5y) 3 = (b 2 + 5y)((b 2 ) 2 – 5yb 2 + (5y) 2 ) = (b 2 + 5y)(b 4 – 5yb 2 + 25y 2 )

7. 5-7 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Solution 2m 5 + 2m 2. Write an equivalent expression by factoring: First factor out the largest common factor: Now factor the sum of two cubes: = 2m 2 (m + 1)(m 2 – m + 1) 2m 5 + 2m 2 = 2m 2 (m 3 + 1).

8. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 5-8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Useful Factoring Facts Sum of cubes: A 3 + B 3 = (A + B)(A 2 – AB + B 2 ) Difference of cubes: A 3 – B 3 = (A – B)(A 2 + AB + B 2 ) Difference of squares: A 2 – B 2 = (A + B)(A – B) There is no formula for factoring a sum of two squares.