SOLUTION: Determine whether the binomial (3x - 2) is a factor of the polynomial {{{3x^3-14x^2+26x-12}}}

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Question 1204305: Determine whether the binomial (3x - 2) is a factor of the polynomial 3x%5E3-14x%5E2%2B26x-12




Found 4 solutions by mananth, greenestamps, ikleyn, math_tutor2020:
Answer by mananth(16946) About Me  (Show Source):
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(3x - 2) is a factor or not
Let 3x-2=0
x=3/2
Plug x=3/2 in the polynomial 3x%5E3-14x%5E2%2B26x-12
f(3/2) = 3*(2/3)^3-14(2/3)^2+26*(2/3)-12 = 0
f(3/2) = 0 therefore 3x-2 is factor of the polynomial

Answer by greenestamps(13200) About Me  (Show Source):
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(3x-2) is a factor of the polynomial if x=2/3 makes the function value zero.

Since no method was specified, here are three....

(1) Graph the function on a graphing calculator and evaluate the function at x=2/3 and see that the function value is 0.

That's fine...but you don't learn any skills by doing it that way.

(2) With pencil and paper, evaluate the function for x=2/3:

3%282%2F3%29%5E3-14%282%2F3%29%5E2%2B26%282%2F3%29-12
3%288%2F27%29-14%284%2F9%29%2B26%282%2F3%29-12
24%2F27-56%2F9%2B52%2F3-12
8%2F9-56%2F9%2B52%2F3-12
-48%2F9%2B52%2F3-12
-16%2F3%2B52%2F3-12
36%2F3-12
12-12
0

Good exercise in arithmetic....

(3) Use synthetic division to verify that x=2/3 is a zero of the function.

      |   3 -14  26 -12
  2/3 |       2  -8  12
      +-----------------
          3 -12  18   0

Synthetic division is a useful skill to have...

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Determine whether the binomial (3x-2) is a factor of the polynomial 3x%5E3-14x%5E2%2B26x-12.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Alternatively to other methods, you can factor the polynomial explicitly,
using the grouping and re-grouping method and applying it step by step.

This is made easier by the fact that the basic binomial 3x-2 is just given to work with it.


3x^3 - 14x^2 + 26x - 12 = group and re-group the terms step by step = (3x^2 - 2x^2) - 12x^2 + 26x - 12 =
= x^2*(3x-2) - (12x^2 -26x) - 12 = x^2*(3x-2) - (12x^2 - 8x) - 8x + 26x - 12 = x^2*(3x-2) - 4x(3x-2) + (18x-12) =
= x^2*(3x-2) - 4x(3x-2) + 6(3x-2) = take off the common factor (3x-2) = (3x-2)*(x^2 - 4x + 6).

The last formula shows explicitly that (3x-2) is the factor.

Solved.



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutors have great solutions.

Here is how it would look like if you used polynomial long division.

Admittedly it's not as efficient as synthetic division.
The fastest method would be to use the remainder theorem to show that f(2/3) = 0.

Because we get remainder 0, it proves that (3x-2) is a factor of 3x^3-14x^2+26x-12

Furthermore, 3x^3-14x^2+26x-12 = (3x-2)(x^2-4x+6)
The quotient is the other factor.
I'll let the student verify this.