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. $32m^3-108n^3$