Question

If a monic cubic polynomial is divided by `x^2-9` leaving a remainder of x+8 and when divided by x leaves a remainder of -4, how do you find this polynomial?

Answer 1

`x^3+4/3x^2-8x-4`

Explanation:

The quotient when a monic cubic polynomial is divided by `x^2-9` must be of the form `x+a`, where `a` is a constant. Thus the polynomial is

`(x+a)(x^2-9)+x+8`

The term independent of `x` is thus `8-9a`. This must be the remainder when the polynomial is divided by `x`, and so

`8-9a=-4 implies a=4/3`

Thus the polynomial is

`(x+4/3)(x^2-9)+x+8`
`=x^3+4/3x^2-8x-4`