SOLUTION: Please help me with the following problem: Solve the problem. Solve the equation 12x³-77x²+91x-30=0 given that 2/3 is a root Thanks in advance

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Question 134150: Please help me with the following problem:
Solve the problem. Solve the equation 12x³-77x²+91x-30=0 given that 2/3 is a root

Thanks in advance
Found 2 solutions by solver91311, josmiceli:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
If you are given that 2/3 is a root, then x=2%2F3 => 3x=2 => 3x-2=0, so 3x-2 is a factor of: 12x%5E3-77x%5E2%2B91x-30

Using either polynomial long division or synthetic division, divide the cubic by the above derived factor. Since the factor was derived from a root, the division will come out even, i.e. no remainder. The quotient will be a quadratic that you can solve either by factoring, completing the square, or use of the quadratic formula.

Let me know if you need any help on any of these steps.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
12x%5E3+-+77x%5E2+%2B+91x+-+30+=+0
Substitute %282%2F3%29%2Ax%5E2+for+x%5E3
Substitute %282%2F3%29%2Ax+for+x%5E2
Substitute 2%2F3+for+x
12x%5E2%2A%282%2F3%29+-+77x%2A%282%2F3%29+%2B+91%2A%282%2F3%29+-+30+=+0
Multiply both sides by 3%2F2
12x%5E2+-+77x+%2B+91+-+45+=+0
12x%5E2+-+77x+%2B+46+=+0
Use quadratiic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%28-77%29+%2B-+sqrt%28+%28-77%29%5E2-4%2A12%2A46+%29%29%2F%282%2A12%29+
x+=+%2877+%2B-+sqrt%283721%29%29%2F24+
x+=+%2877+%2B-+61%29+%2F+24
x+=+19%2F12
x+=+2%2F3
The roots are x+=+19%2F12, x+=+2%2F3, x+=+2%2F3
x+=+2%2F3 is a double root.