Question

Polynomials bx² + x + 5 and bx³ $-$2x + 5 are divided by polynomial x$-$3 and the remainders are m and n respectively. If m$-$ n = 0 then find the value of b.

Answer 1

Let $p\left(x\right)=b{x}^2+x+5$ and $q\left(x\right)=b{x}^3-2x+5$.

The remainder when $p\left(x\right)=b{x}^2+x+5$ is divided by (x − 3) is m.

By remainder theorem,

Remainder = $p\left(3\right)=m$

$$\begin{gathered} \therefore b\times {\left(3\right)}^2+3+5=m \\ \Rightarrow m=9b+8.....\left(1\right) \end{gathered}$$
The remainder when $q\left(x\right)=b{x}^3-2x+5$ is divided by (x − 3) is n.

By remainder theorem,

Remainder = $q\left(3\right)=n$

$$\begin{gathered} \therefore b\times {\left(3\right)}^3-2\times 3+5=n \\ \Rightarrow n=27b-6+5 \\ \Rightarrow n=27b-1.....\left(2\right) \end{gathered}$$
Now,
$$\begin{gathered} m-n=0 \\ \Rightarrow \left(9b+8\right)-\left(27b-1\right)=0\left[\mathrm{Using}\left(1\right)\mathrm{and}\left(2\right)\right] \\ \Rightarrow 9b-27b+8+1=0 \\ \Rightarrow -18b+9=0 \\ \Rightarrow -18b=-9 \\ \Rightarrow b=\frac{-9}{-18}=\frac{1}{2} \end{gathered}$$
Thus, the value of b is $\frac{1}{2}$.