Question

Factorize $x^3+3x^2-x-3$

Answer 1

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

p(x) is a cubic polynomial, so it may have three linear factors.

The constant term is -3. The factors of -3 are -1, 1, -3 and 3.

$p(-1)=(-1{)}^3+3(-1{)}^2-(-1)-3=-1+3+1-3=0$

$\therefore (x+1)$ is a factor of p(x).

$p\left(1\right)=(1{)}^3+3(1{)}^2-1-3=1+3-1-3=0$

$\therefore (x-1)$ is a factor of p(x).

$p(-3)=(-3{)}^3+3(-3{)}^2-(-3)-3=-27+27+3-3=0$

$\therefore (x+3)$ is a factor of p(x).

The three factors of p(x) are (x + 1), (x - 1) and (x - 3)

$\therefore x^3+3x^2-x-3=(x+1)(x-1)(x+3)$.