Question

Let A and B be the remainders when the polynomials $y^3+2y^2-5ay-7$ and $y^3+a{y}^2-12y+6$ are divided by $y+1$ and $y-2$ respectively. If $2A+B=6$, find the value of $a$.

Answer 1

$y^3+2y^2-5ay-7$ leaves a remainder A, when divided by $y+1$.

So, plugging in $y=-1$ in the above polynomial, we can find remainder A.

$A=-1+2+5a-7=5a-6$

$y^3+2y^2-5ay-7$ leaves a remainder B, when divided by $y-2$.

So, plugging in $y=2$ in the above polynomial, we can find remainder A.

$B=8+4a-24+6=4a-10$

Given, $2A+B=6$

$10a-12+4a-10=6$

$14a-22=6$

$14a=28$

$a=2$