Chapter 7: Factoring

# 7.6 Factoring Quadratics of Increasing Difficulty

Factoring equations that are more difficult involves factoring equations and then checking the answers to see if they can be factored again.

Example 7.6.1

Factor $y^4 - 81x^4$.

This is a standard difference of squares that can be rewritten as $(y^2)^2 - (9x^2)^2$, which factors to $(y^2 - 9x^2)(y^2 + 9x^2)$. This is not completely factored yet, since $(y^2 - 9x^2)$ can be factored once more to give $(y - 3x)(y + 3x)$.

Therefore, $y^4 - 81x^4 = (y^2 + 9x^2)(y - 3x)(y + 3x)$.

This multiple factoring of an equation is also common in mixing differences of squares with differences of cubes.

Example 7.6.2

Factor $x^6 - 64y^6$.This is a standard difference of squares that can be rewritten as $(x^3)^2 + (8x^3)^2$, which factors to $(x^3 - 8y^3)(x^3 + 8x^3)$. This is not completely factored yet, since both $(x^3 - 8y^3)$ and $(x^3 + 8x^3)$ can be factored again.

$(x^3-8y^3)=(x-2y)(x^2+2xy+y^2)$ and
$(x^3+8y^3)=(x+2y)(x^2-2xy+y^2)$

This means that the complete factorization for this is:

$x^6 - 64y^6 = (x - 2y)(x^2 + 2xy + y^2)(x + 2y)(x^2 - 2xy + y^2)$

Example 7.6.3

A more challenging equation to factor looks like $x^6 + 64y^6$. This is not an equation that can be put in the factorable form of a difference of squares. However, it can be put in the form of a sum of cubes.

$x^6 + 64y^6 = (x^2)^3 + (4y^2)^3$

In this form, $(x^2)^3+(4y^2)^3$ factors to $(x^2+4y^2)(x^4+4x^2y^2+64y^4)$.

Therefore, $x^6 + 64y^6 = (x^2 + 4y^2)(x^4 + 4x^2y^2 + 64y^4)$.

Example 7.6.4

Consider encountering a sum and difference of squares question. These can be factored as follows: $(a + b)^2 - (2a - 3b)^2$ factors as a standard difference of squares as shown below:

$(a+b)^2-(2a-3b)^2=[(a+b)-(2a-3b)][(a+b)+(2a-3b)]$

Simplifying inside the brackets yields:

$[a + b - 2a + 3b] [a + b + 2a - 3b]$

Which reduces to:

$[-a + 4b] [3a - 2b]$

Therefore:

$(a + b)^2 - (2a - 3b)^2 = [-a - 4b] [3a - 2b]$

Examples 7.6.5

Consider encountering the following difference of cubes question. This can be factored as follows:

$(a + b)^3 - (2a - 3b)^3$ factors as a standard difference of squares as shown below:

$(a+b)^3-(2a-3b)^3$
$=[(a+b)-(2a+3b)][(a+b)^2+(a+b)(2a+3b)+(2a+3b)^2]$

Simplifying inside the brackets yields:

$[a+b-2a-3b][a^2+2ab+b^2+2a^2+5ab+3b^2+4a^2+12ab+9b^2]$

Sorting and combining all similar terms yields:

$\begin{array}{rrl} &[\phantom{-1}a+\phantom{0}b]&[\phantom{0}a^2+\phantom{0}2ab+\phantom{00}b^2] \\ &[-2a-3b]&[2a^2+\phantom{0}5ab+\phantom{0}3b^2] \\ +&&[4a^2+12ab+\phantom{0}9b^2] \\ \hline &[-a-2b]&[7a^2+19ab+13b^2] \end{array}$

Therefore, the result is:

$(a + b)^3 - (2a - 3b)^3 = [-a - 2b] [7a^2 + 19ab + 13b^2]$

# Questions

Completely factor the following equations.

1. $x^4-16y^4$
2. $16x^4-81y^4$
3. $x^4-256y^4$
4. $625x^4-81y^4$
5. $81x^4-16y^4$
6. $x^4-81y^4$
7. $625x^4-256y^4$
8. $x^4-81y^4$
9. $x^6-y^6$
10. $x^6+y^6$
11. $x^6-64y^6$
12. $64x^6+y^6$
13. $729x^6-y^6$
14. $729x^6+y^6$
15. $729x^6+64y^6$
16. $64x^6-15625y^6$
17. $(a+b)^2-(c-d)^2$
18. $(a+2b)^2-(3a-4b)^2$
19. $(a+3b)^2-(2c-d)^2$
20. $(3a+b)^2-(a-b)^2$
21. $(a+b)^3-(c-d)^3$
22. $(a+3b)^3+(4a-b)^3$