Question

If$f\left(x\right)=x^4-2x^3+3x^2-ax-b$ when divided by 𝑥−1,
the remainder is 6, then find the value of 𝑎+𝑏.

Answer 1

Given:$f\left(x\right)=x^4-2x^3+3x^2-ax-b$

We know that if 𝑓(𝑥) is divided by polynomial
(𝑥+𝑎) then by factor theorem remainder
=𝑓(−𝑎).

$\therefore Remainder=f\left(1\right)=6$

$\Rightarrow (1{)}^4-2\times (1{)}^3+3\times (1{)}^2-a\times 1-b=6$

$\Rightarrow 1-2+3-a-b=6$

$\Rightarrow -a-b=4$

$\Rightarrow a+b=-4$

Hence, the value of 𝑎+𝑏 is −4.