Factoring

Factoring is an essential part of any mathematical toolbox. To factor, or to break an expression into factors, is to write the expression (often an integer or polynomial) as a product of different terms. This often allows one to find information about an expression that was not otherwise obvious.

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

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
Simon's Favorite Factoring Trick
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 $(x-y)^3 + (y-z)^3 + (z-x)^3$ .
Factor $x^4 + 1$ into two polynomials with real coefficients

Other Resources

More Common Factorizations.

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