Question

ax^3 -7x^2 + 7x-2, x^3 -2ax^2 +8x-8. I’m to solve for a, and know that when both equations are divided by x-2 they leave the same remainder. So far I’ve verified through synthetic division that they do indeed leave the same remainder, not helpful, but it’s what I have so far. Some helpful insight would be appreciated.

Answer 1

Maybe an easier way to do it is to apply the polynomial remainder theorem. It says that dividing a polynomial
`p(x)` by
`x-c` leaves a remainder of
`p(c)`.

In this case,
`c=2`, and we have

`a(2)^3-7(2)^2+7(2)-2=8a-16`

`(2)^3-2a(2)^2+8(2)-8=16-8a`

Then

`8a-16=16-8a\implies16a=32\implies a=2`