Question

The polynomial f(x) = x4 – 2x3 + 3x2 – ax + b when divided by (x – 1) and (x + 1) leaves the
remainders 5 and 9 respectively. Find the values of a and b.

Answer 1

Dear Student,

$$\begin{gathered} f\left(x\right)=x^4-2x^3+3x^2-ax+b \\ Accordingtothequestion: \\ Whenf\left(x\right)isdividedby(x-1)itleavesaremainderof5. \\ \Rightarrow f\left(1\right)=5 \\ \Rightarrow (1{)}^4-2(1{)}^3+3(1{)}^2-a(1)+b=5 \\ \Rightarrow 1-2+3-a+b=5 \\ \Rightarrow 2-a+b=5 \\ \Rightarrow -a+b=5-2 \\ \Rightarrow -a+b=3..(1) \\ Also, \\ Whenf(x)isdividedby(x+1)itleavesaremainderof9. \\ \Rightarrow f(-1)=9 \\ \Rightarrow (-1{)}^4-2(-1{)}^3+3(-1{)}^2-a(-1)+b=9 \\ \Rightarrow 1+2+3+a+b=9 \\ \Rightarrow 6+a+b=9 \\ \Rightarrow a+b=9-6 \\ \Rightarrow a+b=3..\left(2\right) \\ Adding\left(1\right)and\left(2\right),weget \\ 2b=6 \\ \Rightarrow b=\frac{6}{2} \\ \Rightarrow b=3 \\ Substitutingb=3in\left(2\right) \\ a+b=3 \\ \Rightarrow a+3=3 \\ \Rightarrow a=3-3 \\ \Rightarrow a=0 \\ \mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathbf{:}\mathbf{}\mathit{a}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{b}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3} \end{gathered}$$

Regards