Question

If x³ + x² − ax + b is divisible by (x² − x), write the values of a and b.

Answer 1

Equating x² − x to 0 to find the zeros, we will get
$$\begin{gathered} x\left(x-1\right)=0 \\ \Rightarrow x=0\mathrm{or}x-1=0 \\ \Rightarrow x=0\mathrm{or}x=1 \end{gathered}$$

Since, x³ + x² − ax + b is divisible by x² − x.
Hence, the zeros of x² − x will satisfy x³ + x² − ax + b
$$\begin{gathered} \therefore {\left(0\right)}^3+0^2-a\left(0\right)+b=0 \\ \Rightarrow b=0 \end{gathered}$$
and
$$\begin{gathered} {\left(1\right)}^3+1^2-a\left(1\right)+0=0\left[\because b=0\right] \\ \Rightarrow a=2 \end{gathered}$$