Question

How do you use synthetic division to find the factors of `f(x)= x^4 -x^3 -19x^2+49x-30`?

Answer 1

If you can spot a factor of `f(x)` then you can use synthetic division to divide by that factor to give a simpler polynomial to factor.

Hence we can find:

`f(x) = (x-1)(x+5)(x-2)(x-3)`

Explanation:

First note that `f(1) = 1 - 1 - 19 + 49 - 30 = 0`, so `(x - 1)` is a factor of `f(x)`

Use synthetic division to divide `f(x)` by `(x-1)`:

So `f(x) = (x-1)(x^3-19x+30)`

Notice that `(-5)^3-19*(-5)+30 = -125+95+30 = 0`, so `(x+5)` is also a factor of `f(x)`.

Divide `x^3-19x+30` by `(x+5)` using synthetic division - not forgetting to specify the coefficient `0` of the `x^2` term:

So `f(x) = (x-1)(x+5)(x^2-5+6)`

By this stage you can probably spot that `x^2-5+6 = (x-2)(x-3)` to complete our factorisation:

`f(x) = (x-1)(x+5)(x-2)(x-3)`