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The remainder theorem

When you divide the polynomial $P(x)$ by $x - a$ , the remainder is the number $P(a)$ . You do not need to carry out the whole division to find the remainder. If $P(a) = 0$ , the division is exact and $x - a$ is a factor. Integer zeros of a polynomial with integer coefficients and leading coefficient 1 must divide the constant term, so those are the values you try first.

P(x) = (x - a)Q(x) + P(a)

the remainder is

P(a)

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

a zero gives a factor

Symbols

$a$	the number you divide by, in $x - a$
$P(a)$	the remainder

Example

$P(x) = x^3 + 2x - 5$ divided by $x - 1$ : the remainder is $P(1) = 1 + 2 - 5 = -2$ .

Evaluate

P(a)

directly. Then you skip the whole division when you only need the remainder.

Practise factorisation and polynomial division for free →

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Part of Mathematics 1T: Factorisation and polynomial division.