Chapter 7: Factoring

7.5 Factoring Special Products

Now transition from multiplying special products to factoring special products. If you can recognize them, you can save a lot of time. The following is a list of these special products (note that a2 + b2 cannot be factored):

\begin{array}{lll} a^2-b^2&=&(a+b)(a-b) \\ (a+b)^2&=&a^2+2ab+b^2 \\ (a-b)^2&=&a^2-2ab+b^2 \\ a^3-b^3&=&(a-b)(a^2+ab+b^2) \\ a^3+b^3&=&(a+b)(a^2-ab+b^2) \\ \end{array}

The challenge is therefore in recognizing the special product.

Example 7.5.1

Factor x^2 - 36.

This is a difference of squares. (x - 6)(x + 6) is the solution.

Example 7.5.2

Factor x^2 - 6x + 9.

This is a perfect square. (x - 3)(x - 3) or (x - 3)^2 is the solution.

Example 7.5.3

Factor x^2 + 6x + 9.

This is a perfect square. (x + 3)(x + 3) or (x + 3)^2 is the solution.

Example 7.5.4

Factor 4x^2 + 20xy + 25y^2.

This is a perfect square. (2x + 5y)(2x + 5y) or (2x + 5y)^2 is the solution.

Example 7.5.5

Factor m^3 - 27.

This is a difference of cubes. (m - 3)(m^2 + 3m + 9) is the solution.

Example 7.5.6

Factor 125p^3 + 8r^3.

This is a difference of cubes. (5p + 2r)(25p^2 - 10pr + 4r^2) is the solution.

Questions

Factor each of the following polynomials.

  1. r^2-16
  2. x^2-9
  3. v^2-25
  4. x^2-1
  5. p^2-4
  6. 4v^2-1
  7. 3x^2-27
  8. 5n^2-20
  9. 16x^2-36
  10. 125x^2+45y^2
  11. a^2-2a+1
  12. k^2+4k+4
  13. x^2+6x+9
  14. n^2-8n+16
  15. 25p^2-10p+1
  16. x^2+2x+1
  17. 25a^2+30ab+9b^2
  18. x^2+8xy+16y^2
  19. 8x^2-24xy+18y^2
  20. 20x^2+20xy+5y^2
  21. 8-m^3
  22. x^3+64
  23. x^3-64
  24. x^3+8
  25. 216-u^3
  26. 125x^3-216
  27. 125a^3-64
  28. 64x^3-27
  29. 64x^3+27y^3
  30. 32m^3-108n^3

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-7-5/”>Answer Key 7.5

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