How do you use synthetic division, substitution, and factoring to solve these problems?

Question

1) x^3 - 7x^2 + 14x - 8 = 0


2) -6x^4 + 54x^2 = 0

Answer 1

I honestly don't know for the first one.

For the second one though:

-6x^4 + 54x^2 = 0
-6x^2(x^2 - 9) = 0
-6x^2(x + 3)(x - 3) = 0

x = {-3, 0, 3}

Answer 2

1. The first thing I try on these is, what happens if x = 1 or x = 2? In this case, x = 1 is a root, so if you do a synthetic division of the polynomial by (x-1), you get a quadratic which can be solved by any of the usual methods. I don't want to go into the details of how to do synthetic division here -- it's long, boring, not difficult, and adequately addressed in textbooks.

2. By inspection, 6x^2 is a factor, so the thing has a double root at x = 0. Dividing that out, we are left (after a sign change for convenience) with x^2 -9 = 0, which has roots at x = 3 and x = -3.

Answer 3

x^3 + 2x^2 - 5x = 10 x^3 + 2x^2 - 5x - 10 =0 (x^2-5)(x+2) = 0 ---> x = -sqrt5, sqrt5, -2 x^4 - 17x^2 =18 x^4 - 17x^2 -18=0 (x^2+a million)(x^2-18) = 0 x = -i, i , -3sqrt2, 3sqrt2