Question

For what value of `a` is the polynomial `x^3-ax^2-2x+5a-7` a multiple of `(x-a)`?

Answer 1

The polynomial `x^3-ax^2-2x+5a-7` is a multiple of `(x-a)` for `a=7/3`.

Explanation:

Let `f(x)=x^3-ax^2-2x+5a-7`. `f` is a multiple of `(x-a)` if and only if `f(a)=0` because by the Remainder Theorem, `f(a)` is the remainder after dividing `f` by `(x-a)` and setting the remainder equal to 0 means `(x-a)` is a factor of `f`.

Using synthetic division (or polynomial long division, your choice), we get that the remainder after dividing `x^3-ax^2-2x+5a-7` by `(x-a)` is `3a-7`. We know that `(x-a)` is a factor of `x^3-ax^2-2x+5a-7` if and only if the remainder is 0, so this gives us `3a-7=0` or `a=7/3`.