Question

If the polynomial $\left({\mathrm{x}}^4+2{\mathrm{x}}^3+8{\mathrm{x}}^2+12\mathrm{x}+18\right)$ is divided by another polynomial $\left({\mathrm{x}}^2+5\right)$, the remainder comes out to be $\left(\mathrm{px}+\mathrm{q}\right)$. Find the value of $\mathrm{p}\mathrm{and}\mathrm{q}$

Answer 1

Solution:

Step$\mathbf{1}\mathbf{:}$ Divide $\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{}\mathbf{by}\mathbf{}\mathbf{s}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$:

Given:

$$\begin{gathered} \mathrm{Let}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^4+2{\mathrm{x}}^3+8{\mathrm{x}}^2+12\mathrm{x}+18 \\ \mathrm{Let}\mathrm{s}\left(\mathrm{x}\right)={\mathrm{x}}^2+5 \\ \mathrm{remainder}=\mathrm{px}+\mathrm{q} \end{gathered}$$

$$\begin{gathered} {\mathrm{x}}^2+5\overline{){\mathrm{x}}^4+2{\mathrm{x}}^3+8{\mathrm{x}}^2+12\mathrm{x}+18}\overline{){\mathrm{x}}^2+2\mathrm{x}+3} \\ {\mathrm{x}}^4+5{\mathrm{x}}^2 \\ \overline{2{\mathrm{x}}^3+3{\mathrm{x}}^2+12\mathrm{x}+18} \\ 2{\mathrm{x}}^3+10\mathrm{x} \\ \overline{3{\mathrm{x}}^2+2\mathrm{x}+18} \\ 3{\mathrm{x}}^2+15 \\ \overline{\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{2}\mathbf{x}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{+}\mathbf{3}\mathbf{=}\mathbf{}\mathbf{}\mathbf{Remainder}}\mathbf{} \end{gathered}$$

Step$\mathbf{2}\mathbf{:}$ Compare the remainder:

$$\begin{gathered} \mathrm{px}+\mathrm{q}=2\mathrm{x}+3 \\ \therefore \mathbf{p}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}(\mathrm{on}\mathrm{comparing}\mathrm{with}\mathrm{px}+\mathrm{q}) \\ \mathbf{q}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3} \end{gathered}$$

Final answer: Hence, the value of $\mathrm{p}=\mathbf{2}\mathbf{}\mathbf{and}\mathbf{}\mathrm{q}=\mathbf{3}$.