Factoring

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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:

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

In addition, if $n$ 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,

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

Another way to discover these factorizations is the following: the expression $a^n - b^n$ is equal to zero if $a = b$ . If one factorizes a product which is equal to zero, one of the factors must be equal to zero, so $a^n - b^n$ must have a factor of $a - b$ . Similarly, we note that the expression $a^n + b^n$ when $n$ is odd is equal to zero if $a = -b$ , so it must have a factor of $a - (-b) = a + b$ . Note that when $n$ is even, $(-b)^n + b^n = 2b^n$ , rather than 0, so this gives us no useful information.

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.

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

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

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

Other Useful Factorizations

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.
Given that $a+b+c=0$ , prove that $abc=\dfrac{a^3+b^3+c^3}{3}$ .