Factoring Methods/Tricks

Question

One of the things I've struggled with most in algebra/calculus is all the "factoring tricks". When I take time away from doing math I inevitably forget most if not all of them. The old proverb "use it or lose it" definitely comes into play.

I'm a huge fan of Khan Academy and have started to keep a list of tutorials which demonstrate factoring, but I haven't found them all:

• Factoring and the Distributive Property (3 lessons)
• Solving a quadratic by factoring
• Factoring Trinomials by Grouping (6 lessons)
• Solving Quadratic Equations by Factoring (3 lessons)
• Factoring Special Products (3 lessons)
• Completing the Square
• Factoring perfect square trinomials

Was wondering if anyone could help me fill in the gaps by providing their own list of factoring tips be them in the thread itself, or to online resources like Khan Academy, this site, or elsewhere

Answer 1

As indicated in my comment above, you did not really say what material is covered in your links, but in any case I suggest you look at the algebra section of alcumus:

http://www.artofproblemsolving.com/Alcumus/Introduction.php

Answer 2

Remembering the geometric series helps for factoring things like $a^n-b^n$:

$$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1})$$

A similar factorization applies for $a^\ell+b^\ell$ for $\ell$ odd:

$$\begin{align*} a^3+b^3&=(a+b)(a^2-ab+b^2)\\ a^5+b^5&=(a+b)(a^4-a^3 b+a^2 b^2-ab^3+b^4) \end{align*}$$

whose general pattern I'll leave for you to tease out.

Answer 3

Completing the rectangle:

$$ab+7a+11b=ab+7a+11b+77-77=(a+11)(b+7)-77.$$

Answer 4

Completing a square with a middle term that is a square itself:

$$x^4+4=x^4+4+4x^2-4x^2=(x^2+2)^2-(2x)^2=(x^2+2-2x)(x^2+2+2x).$$