Remainder Theorem

The Remainder theorem formula is used to find the remainder when a polynomial is divided by a linear polynomial.

Remainder Theorem

The Remainder theorem states that "when a polynomial $p(x)$is divided by a linear polynomial $(x-a)$ , then the remainder is $p(a)$"

Example

Find the remainder when the polynomial $p(x)=3x^{3}+x^{2}+2x+5$ is divided by $(x-3)$.

	$3x^{2}-2x+4$
$x+1$	$3x^{3}$	$+$	$x^{2}$	$+$	$2x$	$+$	$5$
	$3x^{3}$	$+$	$3x^{2}$
		$-$	$2x^{2}$	$+$	$2x$	$+$	$5$
		$-$	$2x^{2}$	$-$	$2x$
				$+$	$4x$	$+$	$5$
				$+$	$4x$	$+$	$4$
							1

Here, quotient = $3x^{2}-2x+4$

Remainder = $1$

Verification :

Given, the divisor is $x+1$, i.e. it is a factor of the given polynomial $p(x)$

Let $x+1=0$

$x=-1$

Substituting $x=-1$ in $p(x)$,

$p(x)=3x^{3}+x^{2}+2x+5$

$p(-1-)=3(-1)^{3}+(-1)^{2}+2(-1)+5$

$p(-1-)=3(-1)+1-2+5$

$p(-1-)=-3+1-2+5$

$p(-1-)=1$

p(-1) = 3(-1)3 + (-1)2 + 2(-1) + 5

= 3(-1) + 1 – 2 + 5

= -3 + 4

= 1

Remainder  = Value of p(x) at x = -1.

Hence proved the remainder theorem.

Alternatively,