The idea

Polynomials and the factor theorem

The factor theorem — why f(a) = 0 means (x − a) is a factor — and how to use it with algebraic division to factorise and solve cubics.

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Pure · Algebra & functions

The factor theorem — why f(a) = 0 means (x − a) is a factor — and how to use it with algebraic division to factorise and solve cubics.

A quadratic you can usually factorise by inspection. A cubic like

x^3 - 3x^2 + 4

is harder — you need a way to find a first factor before the rest falls out. That's what the factor theorem gives you.

Divide a polynomial

f(x)

(x - a)

. Whatever happens, you can write

f(x) = (x - a)\,q(x) + r,

where

q(x)

is the quotient and

r

is a constant remainder. Now substitute

x = a

: the

(x-a)

term becomes zero and takes

q(x)

with it, leaving

f(a) = r.

So the value $f(a)$ is the remainder. And

(x - a)

divides exactly — i.e. is a factor — precisely when that remainder is zero:

(x - a) \text{ is a factor of } f(x) \iff f(a) = 0

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