Question

Factorize:$x^3+13x^2+32x+20$.

Answer 1

Step 1: Find a factor of the given polynomial

Let $f\left(x\right)=x^3+13x^2+32x+20$

Factors of $20$ are $\pm 1,\pm 2,\pm 4,\pm 5,\pm 10,\pm 20$

Put $x=-1$

$$\begin{gathered} \Rightarrow f(-1)={(-1)}^3+13{(-1)}^2+32(-1)+20 \\ \Rightarrow f(-1)=-1+13-32+20 \\ \Rightarrow f(-1)=0 \end{gathered}$$

So, $x+1$is the factor of $f\left(x\right)=x^3+13x^2+32x+20$

Step 2: Factorize the given polynomial

Since we obtained $x+1$as one of the factors, we should regroup the terms of given polynomial accordingly.

$$\begin{gathered} f\left(x\right)=x^3+13x^2+32x+20 \\ =x^3+x^2+12x^2+12x+20x+20 \\ =x^2(x+1)+12x(x+1)+20(x+1) \\ =(x+1)(x^2+12x+20) \\ =(x+1)(x^2+10x+2x+20) \\ =(x+1)\left[x(x+10)+2(x+10)\right] \\ =(x+1)(x+10)(x+2) \end{gathered}$$

Hence, the factorized form of the polynomial $x^3+13x^2+32x+20$ is $(x+1)(x+2)$$(x+10)$.