Question

If (x + 1) and (x − 1) are factors of p(x) = ax³ + x² − 2x + b, find the values of a and b.

Answer 1

Let:
$p\left(x\right)=a{x}^3+x^2-2x+b$
Now,
$x=1=0\Rightarrow x=-1$
$$\begin{gathered} \left(x+1\right)\text{is a factor of}p\left(x\right). \\ \Rightarrow p\left(-1\right)=0 \\ \Rightarrow a\times {\left(-1\right)}^3+{\left(-1\right)}^2-2\times \left(-1\right)+b=0 \\ \Rightarrow -a+1+2+b=0 \\ \Rightarrow -a+3+b=0 \\ \Rightarrow a-b=3...\left(\mathrm{i}\right) \end{gathered}$$
Also,
$x-1=0\Rightarrow x=1$
$$\begin{gathered} \left(x-1\right)\text{is a factor of}p\left(x\right). \\ \Rightarrow p\left(1\right)=0 \\ \Rightarrow a\times 1^3+1^2-2\times 1+b \\ \Rightarrow a+1-2+b=0 \\ \Rightarrow a+b-1=0 \\ \Rightarrow a+b=1...\left(\mathrm{ii}\right) \end{gathered}$$
$$\begin{gathered} \mathrm{From}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get}: \\ 2a=4 \\ \Rightarrow a=2 \end{gathered}$$
By substituting the value of a, we get the value of b, i.e., −1.
∴ a = 2 and b = −1