Question

Factorise $x^3+12x^2-31x-42$.

• A. $(x+1)(x-3)(x+14)$
• B. $(x-1)(x-3)(x+14)$
• C. $(x+1)(x+3)(x+14)$
• D. None of these

Answer 1

The correct option is A $(x+1)(x-3)(x+14)$

Let $p\left(x\right)=x^3+12x^2-31x-42$

$p\left(-1\right)=-1+12+31-42=43-43=0$

so by factor theorem

$x+1$ is a factor of $p\left(x\right)$

now, on dividing $p\left(x\right)$ by $x+1$ we get

$p\left(x\right)=\left(x+1\right)\left(x^2+11x-42\right)$

$=\left(x+1\right)\left(x^2+14x-3x-42\right)$

$=\left(x+1\right)\left(x\left(x+14\right)-3\left(x+14\right)\right)$

=$\left(x+1\right)\left(x-3\right)\left(x+14\right)$