A polynomial f and a factor of f are given. factor f completely.?

Question

f(x)= 6x^5 - 38x^4 + 12x^3 - 15x^2 + 95x - 30; x-6

Answer 1

Synthetic division:

+06 | +06 −38 +12 −15 +95 −30
−−−| +00 +36 −12 +00 −90 +30
−−−−−−−−−−−−−−−−−−−−−−−
−−−| +06 −02 +00 −15 +05 +00

so you're left with:

6x^4 - 2x^3 - 15x + 5

Again note that the coefficients have the same ratio, so you can factor:

2x³(3x - 1) - 5(3x - 1)
(3x - 1)(2x³ - 5)

Then you can still factor the 2x³ - 5, though it's really ugly, using the difference of cubes rule:

2x³ - 5 = (³√2x - ³√5)(³√4x² + ³√100x + ³√25)

So all together, you'd get:

(x - 6)(3x - 1)(³√2x - ³√5)(³√4x² + ³√100x + ³√25)

Answer 2

Divide to cut back the cubic to a quadratic. synthetic branch: 3 | 2 -3 -8 -3 ....2  3  a million 0 = 2x^2 + 3x + a million which by opposite FOIL is (2x + a million)(x + a million) So (2x + a million)(x + a million)(x - 3)