Question

Use synthetic division method for performing following divisions. Write whether the divisor is a factor of dividend. Justify.

(i) (3p⁴ $-$ 4p³ $-$ 3p $-$ 1) $÷$ (p $-$ 1)

Answer 1

$$\begin{gathered} \mathrm{Dividend}=3p^4-4p^3-3p-1 \\ \mathrm{Index}\mathrm{form}=3p^4-4p^3+0p^2-3p-1 \\ \mathrm{Coefficient}\mathrm{form}=\left(3,-4,0,-3,-1\right) \\ \mathrm{Comparing}\mathrm{the}\mathrm{divisor}p-1\mathrm{with}p-a,\mathrm{we}\mathrm{get}a=1 \end{gathered}$$



$$\begin{gathered} \therefore \mathrm{Quotient}\mathrm{in}\mathrm{coefficient}\mathrm{form}=\left(3,-1,-1,-4\right) \\ \mathrm{i}.\mathrm{e}.,\mathrm{quotient}=3p^3-p^2-p-4\mathrm{in}\mathrm{the}\mathrm{variable}p \\ \mathrm{Here},\mathrm{the}\mathrm{remainder}\mathrm{is}-5\left(\mathrm{nonzero}\right). \\ \therefore \left(p-1\right)\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{factor}\mathrm{of}\left(3p^4-4p^3-3p-1\right). \end{gathered}$$