1. ## Question on polynomials:

If $\displaystyle P(x)$ has remainder $\displaystyle 2$ when divided by $\displaystyle (x-3)$ and remainder $\displaystyle -13$ when divided by $\displaystyle (x+2)$, find the remainder when $\displaystyle P(x)$ is divided by $\displaystyle (x^2 - x - 6)$

I know that $\displaystyle P(k)=Q(k)(k-k)+R =R$

Please, I need help for this one!

2. Originally Posted by karldiesen
If $\displaystyle P(x)$ has remainder $\displaystyle 2$ when divided by $\displaystyle (x-3)$ and remainder $\displaystyle -13$ when divided by $\displaystyle (x+2)$, find the remainder when $\displaystyle P(x)$ is divided by $\displaystyle (x^2 - x - 6)$

I know that $\displaystyle P(k)=Q(k)(k-k)+R =R$

Please, I need help for this one!
So you know that P(x)= (x-3)Q(x)+ 2= (x+2)R(x)- 13. Then (x+2)P(x)= (x+2)(x-3)Q(x)+ 2(x+2) and (x-3)P(x)= (x+2)(x-3)R(x)- 13(x- 3). Subtracting the two equations, ((x+2)- (x- 3))P(x)= 5P(x)= (x+2)(x-3)(Q(x)- R(x))+ 15x- 35. P(x)= (x+2)(x-3)S(x)+ 3x- 7 where S(x) is (Q(x)- R(x))/5.