Remainder Theorem

Maths: Edexcel A Level Pure Maths, topic 2.4.3

Remainder Theorem

Dividing polynomials can be a tricky process. The remainder theorem allows us to quickly find the remainder of a polynomial when it is divided by a binomial.

Division

Division

A given polynomial can be decomposed into a quotient, divisor and remainder as follows:
$f(x) = q(x)(x-c) + r$
Where $f(x)$ is the polynomial, $q(x)$ is the quotient, $x-c$ is the divisor and $r$ is the remainder.

Division

Division

When evaluated for the value $c$ , this decomposition can be written as:
$f(c) = q(c)(c-c) + r = r$ .
This leads us to the remainder theorem.

Remainder theorem

Remainder theorem

If the polynomial function $f(x)$ is divided by $x-c$ , then the remainder is $f(c)$ .
We can use this to find the remainder of polynomials for given devisors without actually having to do the division!
$f(c) = 0$ if and only if $x-c$ is a factor of $f(x)$ .

Test yourself on Remainder Theorem

Which part of the following equation is known as the divisor?
$f(x) = q(x)(x-c)+r$
Which part of the following equation is known as the quotient?
$f(x) = q(x)(x-c)+r$
Which part of the following equation is known as the remainder?
$f(x) = q(x)(x-c)+r$

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