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)
(3x-1)(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.
http://en.wikipedia.org/wiki/Rational_root_theorem
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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To factor the polynomial 3x^3-4x^2+4x-1, we can use the method of synthetic division or polynomial long division.
Let's use polynomial long division to factor the given polynomial:
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x-1 | 3x^3 - 4x^2 + 4x - 1
First, we need to focus on the leading term, which is 3x^3, and divide it by the leading term of the divisor, which is x. This gives us 3x^2.
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x-1 | 3x^3 - 4x^2 + 4x - 1
3x^2
Next, we multiply the divisor (x-1) by the quotient we just obtained (3x^2) and subtract it from the original polynomial:
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x-1 | 3x^3 - 4x^2 + 4x - 1
3x^2 - 3x^2
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0 - 4x^2 + 4x - 1
We repeat this process with the new polynomial (0 - 4x^2 + 4x - 1) and the same divisor (x-1).
The leading term of the new polynomial is -4x^2. Dividing it by x gives us -4x:
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x-1 | 3x^3 - 4x^2 + 4x - 1
3x^2 - 3x^2 - 4x
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0 - 4x^2 + 4x - 1
0 + 4x^2 - 4x
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0 - 1
Since the remainder is -1 (a constant term), we now have a quadratic polynomial (0 - 4x^2 + 4x - 1) and can factor it further.
The quadratic polynomial can be factored as (-1)(2x - 1)(2x - 1), where (2x - 1) is a repeated factor. The -1 comes from the remainder, and (2x - 1) comes from the factored terms in the polynomial.
Therefore, the factored form of 3x^3 - 4x^2 + 4x - 1 is:
(3x^2 - 4x + 1)(2x - 1)(2x - 1)