Question

The polynomials (ax³ + 3x² − 3) and (2x³ − 5x + a) when divided by (x − 4) leave the same remainder. Find the value of a.

Answer 1

Let:
$f\left(x\right)=a{x}^3+3x^2-3\mathrm{and}g\left(x\right)=2x^3-5x+a$
Now,
$$\begin{gathered} \mathrm{When}f\left(x\right)\mathrm{is}\mathrm{divided}\mathrm{by}\left(x-4\right),\mathrm{the}\mathrm{remainder}\mathrm{is}f\left(4\right). \\ \mathrm{When}g\left(x\right)\mathrm{is}\mathrm{divided}\mathrm{by}\left(x-4\right),\mathrm{the}\mathrm{remainder}\mathrm{is}g\left(4\right). \end{gathered}$$
We now have:
$$\begin{gathered} f\left(4\right)=\left(a\times 4^3+3\times 4^2-3\right) \\ =\left(64a+48-3\right) \\ =\left(64a+45\right) \end{gathered}$$
And,
$$\begin{gathered} g\left(4\right)=\left(2\times 4^3-5\times 4+a\right) \\ =\left(128-20+a\right) \\ =\left(108+a\right) \end{gathered}$$
Thus, we have:
$$\begin{gathered} 64a+45=108+a \\ \Rightarrow 64a-a=108-45 \\ \Rightarrow 63a=63 \\ \Rightarrow a=1 \end{gathered}$$