Question

If the expression $a{x}^4+b{x}^3-x^2+2x+3$ has remainder $4x+3$ when divided by $x^2+x-2$, then the value of $a$ and $b$, is

• A. $a=1,b=-2$
• B. $a=1,b=2$
• C. $a=-1,b=2$
• D. None of these

Answer 1

Answer: (B)

$a=1,b=2$

As $x^2+x-2=(x+2)\times (x-1)$
Let $f\left(x\right)=a{x}^4+b{x}^3-x^2+2x+3$
Then
$f(-2)=a(-2{)}^4+b(-2{)}^3-(-2{)}^2+2(-2)+3=4(-2)+3$
$\Rightarrow 16a-8b-4-4+3=-5\Rightarrow 2a-b=0\to$ (1)
And
$f\left(1\right)=a+b-1+2+3=4\left(1\right)+3\Rightarrow a+b=3\to$ (2)
From (1) and (2), we get
$a=1,b=2$