Question

The polynomial $f\left(x\right)=x^4-2x^3+3x^2-ax+b$ when divided by $(x-1)$ and $(x+1)$ leaves the remainders $5$ and $19$ respectively. Find the values of $a$ and $b$. Hence, find the remainder when $f\left(x\right)$ is divided by $(x-2)$

• A. $a=6,b=8$ and $remainder=10$
• B. $a=5,b=8$ and $remainder=10$
• C. $a=5,b=7$ and $remainder=10$
• D. None of these

Answer 1

The correct option is B $a=5,b=8$ and $remainder=10$

by remainder theorem,

$f\left(1\right)=5$ $and$ $f(-1)=19$

$\therefore 1-2+3-a+b=5\Rightarrow b-a=3$ and

$1+2+3+a+b=19\Rightarrow a+b=13$

$b=8$ $a=5$

and remainder when $f\left(x\right)$ is divided by $(x-2)$ is

$f\left(2\right)=16-16+12-10+8=10$