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If $x^2+x-6$ is a factor of $2 x^3+x^2+a x+b$, then $6 a+13 b=$
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Verified Answer
The correct answer is:
$0$
Since, $x^2+x-6=0$
$\Rightarrow(x+3)(x-2)=0 \Rightarrow x=-3,2$
So, $2(-3)^3+(-3)^2+a(-3)+b=0$
$\Rightarrow-3 \mathrm{a}+\mathrm{b}=45$ ... (i)
and $2(2)^3+(2)^2+2 a+b=0$
$\Rightarrow 2 \mathrm{a}+\mathrm{b}=-20$ ... (ii)
After solving (i) \& (ii) we get
$a=-13, b=6$
Now, $6 a+13 b=-65+65=0$
$\Rightarrow(x+3)(x-2)=0 \Rightarrow x=-3,2$
So, $2(-3)^3+(-3)^2+a(-3)+b=0$
$\Rightarrow-3 \mathrm{a}+\mathrm{b}=45$ ... (i)
and $2(2)^3+(2)^2+2 a+b=0$
$\Rightarrow 2 \mathrm{a}+\mathrm{b}=-20$ ... (ii)
After solving (i) \& (ii) we get
$a=-13, b=6$
Now, $6 a+13 b=-65+65=0$
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