Question

Factorise $x^3-3x^2-9x-5$

Answer 1

Find the factors

The given polynomial is $x^3-3x^2-9x-5$

The polynomial can be simplified as

$$\begin{gathered} x^3+x^2-4x^2-4x-5x-5 \\ =x^2(x+1)-4x(x+1)-5(x+1) \\ =(x+1)(x^2-4x-5) \\ =(x+1)(x^2-5x+x-5) \\ =(x+1)\left\{x(x-5)+(x-5)\right\} \\ =(x+1)(x-5)(x+1) \\ ={\left(x+1\right)}^2(x-5) \end{gathered}$$

Hence, the factors of the polynomial $x^3-3x^2-9x-5$ are $\left(x+1\right),(x+1)$ and $\left(x-5\right)$