Question

If x+1 and x-1 are the factors of the polynomial ax^3 + x^2 - 2x +b,find the values of a and b

Answer 1

The polynomial remainder theorem says that dividing a polynomial $p(x)$ by $x-c$ leaves a remainder of $p(c)=0$ if $x-c$ is a factor of $p(x)$. In this case, check $c=-1$ and $c=1$.

$a(-1)^3+(-1)^2-2(-1)+b=-a+b+3=0$

$a(1)^3+(1)^2-2(1)+b=a+b-1=0$

From the first equation,

$-a+b+3=0\implies a=b+3$

and substituting into the second gives

$(b+3)+b-1=0\implies2b+2=0\implies b=-1\implies a=2$