Question

Using the Remainder and Factor Theorem, factorise the following polynomial: $x^3+10x^2-37x+26$

Answer 1

Let $f\left(x\right)=x^3+10x^2-37x+26$
Putting $x=1$, we get
$f\left(1\right)=1+10-37+26=0$
$\therefore$ By factor theorem, $x-1$ is factor of $f\left(x\right)$.
On dividing $x^3+10x^2-37x+26$ by $x-1$, we get $x^2+11x-26$ as the quotient and remainder $=0$.
$\therefore$ The other factor of $f\left(x\right)$ are the factor of $x^2+11x-26$
Now, $x^2+11x-26$
$=x^2+13x-2x-26$
$=x(x+13)-2(x+13)$
$=(x+13)(x-2)$
Hence, $x^3+10x^2-37x+26=(x-1)(x-2)(x+13)$