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):

[latex]\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}[/latex]

The challenge is therefore in recognizing the special product.

Example 7.5.1

Factor [latex]x^2 - 36[/latex].

This is a difference of squares. [latex](x - 6)(x + 6)[/latex] is the solution.

Example 7.5.2

Factor [latex]x^2 - 6x + 9[/latex].

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

Example 7.5.3

Factor [latex]x^2 + 6x + 9[/latex].

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

Example 7.5.4

Factor [latex]4x^2 + 20xy + 25y^2[/latex].

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

Example 7.5.5

Factor [latex]m^3 - 27[/latex].

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

Example 7.5.6

Factor [latex]125p^3 + 8r^3[/latex].

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

Questions

Factor each of the following polynomials.

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

Answer Key 7.5

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