Question

If the remainder equals 5 , when 2x³ – 2x² + x - a is divided by x-2, then find the value of a.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{p}\left(\mathrm{x}\right)=2{\mathrm{x}}^3-2{\mathrm{x}}^2+\mathrm{x}-\mathrm{a} \\ \mathrm{Now},\mathrm{by}\mathrm{remainder}\mathrm{theorem},\mathrm{when}\mathrm{p}\left(\mathrm{x}\right)\mathrm{is}\mathrm{divided}\mathrm{by}\left(\mathrm{x}-2\right),\mathrm{then}\mathrm{remainder}\mathrm{is}\mathrm{p}\left(2\right). \\ \mathrm{Now},\mathrm{p}\left(\mathrm{x}\right)=2{\mathrm{x}}^3-2{\mathrm{x}}^2+\mathrm{x}-\mathrm{a} \\ \Rightarrow \mathrm{p}\left(2\right)=2{\left(2\right)}^3-2{\left(2\right)}^2+2-\mathrm{a} \\ \Rightarrow \mathrm{p}\left(2\right)=16-8+2-\mathrm{a} \\ \Rightarrow \mathrm{p}\left(2\right)=10-\mathrm{a} \\ \mathrm{Now},\mathrm{p}\left(2\right)=5 \\ \Rightarrow 10-\mathrm{a}=5 \\ \Rightarrow -\mathrm{a}=5-10 \\ \Rightarrow \boxed{\mathbf{a}\mathbf{}\mathbf{=}\mathbf{}\mathbf{5}} \end{gathered}$$