Common factorization situations

It is quite valuable to know how to factor differences of powers and (when possible) sums of powers:

$$(a^n-b^n)=(a-b)(a^{n-1}+a^{n-2}b+ a^{n-3}b^2 + \ldots + b^{n-1}$$

With the familiar cases

$$(a^2-b^2)=(a-b)(a+b)$$

and

$$(a^3-b^3)=(a-b)(a^2+ab+b^2).$$

For odd powers we also can factor the sum:

$$(a^3+b^3)=(a+b)(a^2-ab+b^2)$$

$$(a^5+b^5)=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)$$

A quadratic $x^2+ax +b$ which is factorable into $(x+r_1)(x+r_2)$ will have $r_1r_2=b$ and $r_1+r_2=a$.