Question 137337
Solve the problem. Solve the equation 12x to third power - 77x to second power + 91x - 30 = 0 given that 2/3 is a root.
<pre>
{{{12x^3 - 77x^2 + 91x - 30 = 0 }}} given that {{{2/3}}} is a root.

Since {{{2/3}}} is a root, then the polynomial is divisible by {{{x-2/3}}}

So we can divide that synthetically this way:

{{{2/3}}} | 12 -77  91 -30
   |      8 -46  30 
    ---------------
     12 -69  45   0

So we have factored the polynomial this way

{{{(x-2/3)(12x^2-69x+45)=0}}} 

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

{{{(x-2/3)3(4x^2-23x+15)=0}}}

Factor the trinomial in the second parentheses
      
{{{(x-2/3)3(x-5)(4x-3)=0}}}

If you like you can move the {{{3}}} factor in front of
the first factor:

{{{3(x-2/3)(x-5)(4x-3)=0}}}

and distribute the 3 into the {{{(x-2/3)}}}

{{{(3x-2)(x-5)(4x-3)=0}}}

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

{{{3x-2=0}}} gives solution {{{x=2/3}}} which we knew from the start.
{{{x-5=0}}} gives solution {{{x=5}}}
{{{4x-3=0}}} gives solution {{{x=3/4}}}

Edwin</pre>