Factoring

Factoring is an essential part of problem solving. It is part of number theory and applying it will yield many results. There are many methods to factor, ranging all the way from using the Distributive Property to make an expression easier to calculate to using factoring as a vital part of proofs.

Differences and Sums of Powers

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

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

Using the formula for the sum of a geometric sequence, it's easy to derive the more general formula:

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

Take note of the specific case where n is odd:

$a^n+b^n=(a+b)(a^{n-1} - ba^{n-2} + b^2a^{n-3} - b^3a^{n-4} + \cdots + b^{n-1})$

This also leads to the formula for the sum of cubes,

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

Vieta's/Newton Factorizations

These factorizations are useful for problems that could otherwise be solved by Newton sums or problems that give a polynomial and ask a question about the roots. Combined with Vieta's formulas, these are excellent factorizations that show up everywhere.

$\displaystyle (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$

$\displaystyle (a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a)$

$\displaystyle (a+b+c)^5=a^5+b^5+c^5+5(a+b)(b+c)(c+a)(a^2+b^2+c^2+ab+bc+ca)$

Other Useful Factorizations

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
See Simon's Favorite Factoring Trick (This is not a recognized formula, please do not quote it on contests)
Binomial theorem

Practice Problems

Prove that $n^2 + 3n + 5$ is never divisible by 121 for any positive integer ${n}$ .
Prove that $2222^{5555} + 5555^{2222}$ is divisible by 7. - USSR Problem Book
Factor $(x-y)^3 + (y-z)^3 + (z-x)^3$ .

Other Resources

More Common Factorizations.

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