12th grade Math?
posted by BioBabe on .
How would you factor 3x^3-4x^2+4x-1?
P.S. Factor theorem does not work here.
Look for the first root by trial and error. Using the "q/p theorem", I see that one solution is +1/3. So, (x - 1/3) is one factor.
Divide that into 3x^3-4x^2+4x-1 to get a quadratic factor,
(3x^2 -3x +3) = 3 (x^2 -x +1).
The term in parentheses can be factored by the usual means, but gives two complex roots.
3x^3-4x^2+4x-1 = 3(x- 1/3)(x^2-x+1)
What's the q/p theorem though?
I was afraid you'd ask me that :-)
The more common name for it is the rational roots test (or theorem).
Here is a reference.
It doesn't always provide a root, but if there is a rational real root, it works.
Briefly, it says that if the constant term of the polynomial is q and the first term in p, and if there are rational real roots, one of the roots will be
+/- q/p or +/- the tio of prime-number factors of q and p.
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