SOLUTION: (b) Solve for x, the equation 12x3 + 8x2 - x - 1 = 0. (All solutions are rational and between ±1.)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: (b) Solve for x, the equation 12x3 + 8x2 - x - 1 = 0. (All solutions are rational and between ±1.)      Log On


   

Question 330385: (b) Solve for x, the equation 12x3 + 8x2 - x - 1 = 0. (All solutions are rational and between ±1.)
Answer by galactus(183) About Me  (Show Source):
You can put this solution on YOUR website!
12x%5E3%2B8x%5E2-x-1=0
Let's try breaking it up, grouping and factoring. One could use the Rational Root Theorem as well.
Rewrite:
12x%5E3%2B12x%5E2-4x%5E2-4x%2B3x-1=0
Group:
12x%5E3%2B12x%5E2%2B3x-4x%5E2-4x-1
Factor out 3x from the first 3 terms and -1 from the last 3 terms:
3x%284x%5E2%2B4x%2B1%29-%284x%5E2%2B4x%2B1%29
%283x-1%29%284x%5E2%2B4x%2B1%29
%283x-1%29%282x%2B1%29%5E2=0
What makes 3x-1=0?. x=1/3
What makes 2x+1=0?. x=-1/2
Since x=-1/2 has multiplicity 2, we have three roots as required of the cubic.