Question

If (x³ + mx² − x + 6) has (x − 2) as a factor and leaves a remainder r when divided by (x − 3), find the values of m and r.

Answer 1

$$\begin{gathered} \mathrm{Let}f\left(x\right)=x^3+m{x}^2-x+6. \\ \mathrm{Here},\left(x-2\right)\mathrm{is}\mathrm{a}\mathrm{factor}\mathrm{of}f\left(\mathrm{x}\right). \\ \mathrm{Therefore},f\left(2\right)=0 \\ \Rightarrow 2^3+m\times 2^2-2+6=0 \\ \Rightarrow 8+4m-2+6=0 \\ \Rightarrow 4m+12=0 \\ \Rightarrow 4m=-12 \\ \Rightarrow m=-3 \\ \mathrm{Now},\mathrm{using}m=-3inf\left(x\right),\text{we get}: \\ f\left(x\right)=x^3-3x^2-x+6 \\ \mathrm{When}f\left(x\right)\text{is}\mathrm{divided}\mathrm{by}\left(x-3\right),\mathrm{the}\mathrm{remainder}\mathrm{will}\mathrm{be}f\left(3\right), \\ \mathrm{and}\text{the}\mathrm{remainder}\mathrm{is}\mathrm{given}\mathrm{as}r. \\ \mathrm{Now},f\left(3\right)=3^3-3\times 3^2-3+6 \\ \mathrm{or},r=27-27-3+6 \\ \mathrm{or},r=3 \\ \mathrm{Therefore},m=-3\mathrm{and}r=3. \end{gathered}$$