Question

Factorize the equation: $x^3-3x^2-9x-5$.

Answer 1

Step 1: Split the middle terms

Let $f\left(x\right)=x^3-3x^2-9x-5$

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

Step 2: Factorize the quadratic equation

$$\begin{gathered} f\left(x\right)=(x+1)(x^2-5x+x-5) \\ f\left(x\right)=(x+1)\left[x(x-5)+1(x-5)\right] \\ f\left(x\right)=(x+1)(x-5)(x+1) \end{gathered}$$

Hence, the factorized form of the polynomial $x^3-3x^2-9x-5$ is $(x+1)(x+1)$$\left(x-5\right)$.