Question

Given that $x^2+x-6$ is a factor of $2x^4+x^3-a{x}^2+bx+a+b-1$, then find the values of $a$ and $b$.

• A. $a=16,b=3$
• B. $a=16,b=2$
• C. $a=10,b=3$
• D. None of these

Answer 1

Answer: (A)

$a=16,b=3$

As $x^2+x-6=(x+3)(x-2)$
Let $f\left(x\right)=2x^4+x^3-a{x}^2+bx+a+b-1$
We get
$f(-3)=2(-3{)}^4+(-3{)}^3-a(-3{)}^2+b(-3)+a+b-1=0$
$\Rightarrow 134-8a-2b=0\Rightarrow 4a+b=67\to$ (1)
And
$f\left(2\right)=2(2{)}^4+2^3-a(2{)}^2+2b+a+b-1=0$
$\Rightarrow 39-3a+3b=0\Rightarrow a-b=13\to$(2)
Solving (1) and (2), we get
$a=16,b=3$