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 which of the following is/are true?

• A. $b=2$
• B. $a=2$
• C. $b=1$
• D. $a=1$

Answer 1

Answer: (A), (D)

$a=1$

Let $f\left(x\right)=a{x}^4+b{x}^3-x^2+2x+3$
Using remainder theorem,
$f\left(x\right)=(x^2+x-2)Q\left(x\right)+4x+3$
$f\left(x\right)=(x+2)(x-1)Q\left(x\right)+4x+3$

$\therefore f(-2)=4(-2)+3=-5$
$\Rightarrow a(-2{)}^4+b(-2{)}^3-(-2{)}^2+2(-2)+3=-8+3$
$\Rightarrow 16a-8b-4-4+3=-5$
$\Rightarrow 16a-8b-5=-5$
$\Rightarrow 16a-8b=0$
$\Rightarrow 2a=b.....\left(i\right)$

And
$f\left(1\right)=4\left(1\right)+3=7$
$\Rightarrow a(1{)}^4+b(1{)}^3-(1{)}^2+2(1)+3=4+3=7$
$\Rightarrow a+b-1+2+3=7$
$\Rightarrow a+b=3.....\left(ii\right)$

On solving $\left(i\right)\text{and}\left(ii\right),$
$2a=b\text{and}a+b=3$,
we get $a=1,b=2$