Factorising with Ruffini's rule

When the remainder of Ruffini’s division is zero, the divisor is a factor of the polynomial: this is why Ruffini’s rule is also a powerful factorisation technique for polynomials of degree $\ge 3$.

Property — Zero of a polynomial

If substituting $x=a$ into the polynomial $P(x)$ gives $P(a)=0$, then $(x-a)$ is a divisor of $P(x)$ and the remainder of Ruffini’s division is zero. The polynomial factorises as: $P(x) = (x-a)\cdot Q(x).$ The number $a$ is called a zero of the polynomial.

Example

Let us factorise $x^3-2x^2-5x+6$.

We try $x=1$: $1-2-5+6=0$ ✓, so $(x-1)$ is a factor.

	$x^3$	$x^2$	$x$	$1$
	$1$	$-2$	$-5$	$6$
$c=1$		$1$	$-1$	$-6$
	$1$	$-1$	$-6$	$\boxed{0}$

Therefore $x^3-2x^2-5x+6 = (x-1)(x^2-x-6) = \boxed{(x-1)(x-3)(x+2)}$.

Links

Topics: Algebraic calculation
Concepts: Ruffini’s rule · Factorisation · Zero of a polynomial
Skills: Factorising
People: Paolo Ruffini