SOLUTION: Solve the problem. Solve the equation 12x to third power - 77x to second power + 91x - 30 = 0 given that 2/3 is a root.

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If is a root of the equation, then is a factor of the cubic polynomial.

Use polynomial long division to divide by . Since is a factor and you are dividing a 1st degree polynomial into a 3rd degree polynomial, the quotient will be a quadratic with no remainder. Solve the quadratic by normal methods to obtain the two missing roots to the original equation.

You can put this solution on YOUR website!
Solve the problem. Solve the equation 12x to third power - 77x to second power + 91x - 30 = 0 given that 2/3 is a root.

 given that  is a root.

Since  is a root, then the polynomial is divisible by 

So we can divide that synthetically this way:

 | 12 -77  91 -30
   |      8 -46  30 
    ---------------
     12 -69  45   0

So we have factored the polynomial this way

 

Now we can factor a 3 out of the second parentheses, getting:



Factor the trinomial in the second parentheses
      


If you like you can move the  factor in front of
the first factor:



and distribute the 3 into the 



That's the complete factorization of the
original polynomial. Now we use the zero-factor
principle and set each factor equal to 0

 gives solution  which we knew from the start.
 gives solution 
 gives solution 

Edwin