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

1 Answer
George C.
Aug 30, 2015

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):
enter image source here
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:
enter image source here
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)

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