Question

If the polynomials az³ + 4z² + 3z − 4 and z³ − 4z + a leave the same remainder when divided by z − 3, find a.

Answer 1

Let $p\left(z\right)=a{z}^3+4z^2+3z-4\mathrm{and}q\left(z\right)=z^3-4z+a$.
Since, p(z) and q(z) leave the same remainder when divided by z − 3, so
$$\begin{gathered} p\left(3\right)=q\left(3\right) \\ \Rightarrow a{\left(3\right)}^3+4{\left(3\right)}^2+3\left(3\right)-4={\left(3\right)}^3-4\left(3\right)+a \\ \Rightarrow 27a+36+9-4=27-12+a \\ \Rightarrow 26a=-26\Rightarrow a=-1 \end{gathered}$$
Hence, a = −1.