Question 937897: Hello!
I am confused on how to factor this 6th degree polynomial:
f(x) = 6x^4-13x^3-27x^2+40x-12
How would I group them? I tried to rearrange them but I can't seem to find a way to figure it out.
Any help is greatly appreciated. Thank you!
- Marterlo
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39618)   (Show Source):
You can put this solution on YOUR website! Maybe this misses some fundamental skills, but maybe try some software, or a program of you own design to apply synthetic division to look for roots, or to test each possible root (to see if the remainder is 0). Rational Roots theorem will help you find several roots to check, and then synthetic division should tell you if the root is part of the function f(x). On paper, the division process can become messy. That's why I suggest some kind of software.
You might also try a graphing tool to indicate rational roots, which can tell you directly the binomial factors of f(x).
Using the graphing feature of Google.com, two roots are at -2 and 3. Another couple roots occur between them; one seems to be 0.5, and then,....
by that extent, you would more easily be able to analyze the simpler factor remaining after the three synthetic divisions.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website! Hello!
I am confused on how to factor this 6th degree polynomial:
f(x) = 6x^4-13x^3-27x^2+40x-12
How would I group them? I tried to rearrange them but I can't seem to find a way to figure it out.
Any help is greatly appreciated. Thank you!
- Marterlo
You can use the RATIONAL ROOT THEOREM twice, and along with POLYNOMIAL LONG DIVISION,
you should obtain the factors of polynomial:  as:  , and so:
 becomes:  , from which the solutions to the equation,
or the ROOTS of the equation can be obtained.
By the way, it's a 4th degree polynomial, not a 6th degree polynomial. Hence, there are 4 ROOTS, or 4 solutions, or 4 zeroes.
You can do the check!!
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