Given polynomial f(x) and a factor of f(x), factor f(x) completely?

Divide f(x) by the given factor:

(x3 - 10x2 + 19x + 30) / (x - 6) = x^2 - 4x - 5

Factor the reduced quadratic:

(x - 5) (x + 1)

So altogether x3 - 10x2 + 19x +30 = (x - 6) (x - 5) (x + 1)

you may in basic terms carry out a tail-branch utilising the polynomials. it fairly is using the actuality all and sundry understand that f(ok) = 0, so f(x) = (x-ok)g(x), with g(x) a polynomial of degree 2. i circulate to objective to place in writing the tail branch as needless to say as feasible: x^3 - 3x^2-16x-12 | x-6 | x is going in x^3 precisely x^2 situations: x^3 - 3x^2-16x-12 | x-6 | x^2 x^3 - 6x^2 (the synthetic from x^2 and (x-6)) subtract: 3x^2-16x-12 | x-6 | x^2 3x^2-16x-12 | x-6 | x^2 +3x 3x^2-18x 2x-12 | x-6 | x^2 + 3x 2x-12 | x-6 | x^2 +3x +2 2x-12 0 with the aid of actuality the proper something is 0, the dept is sweet, and we get that f(x)=x^3-3x^2-16x-12 = (x-6)(x^2 + 3x +2). the 2d element is a quadratic polynomial, which will real be factored into (x-a million)(x-2) (in basic terms use the nicely-known technique for fixing a quadratic equation). the top-results of factoring f(x) = (x-6)(x-a million)(x-2)