Question

For what value of `b` will the polynomial `P(x) = 4x^3-3x^2+bx+6` have the same remainder when it is divided by both `x` and `x+3`?

Answer 1

`b=-45`

Explanation:

The remainder when divided by `x-a` is `P(a)`

So we require `P(0) = P(-3)`

`P(0) = 6`

`P(-3) = 4(-27)-3(9)+b(-3)+6 = -108-27-3b+6 = -3b-129`

So we need:

`6 = P(0) = P(-3) = -3b-129`

Adding `3b-6` to both ends this becomes:

`3b = -135`

Then dividing both sides by `3` we find:

`b = -45`