Factoring 12z^3+z-1
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Please help me factor the polynomial:
12z^3+z-1.
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Solution Summary
We explain how to obtain the complete factorization of this polynomial.
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If we use the Rational Roots Theorem we find the following possible rational roots:
z = +/- 1, +/- 1/2, +/- 1/3, +/- 1/4, +/- 1/6, +/- 1/12
We find that none of these are zeroes. So we need to factor the polynomial using the standard solution of the cubic equation. You can easily derive this as follows for this case. The identity:
(a+b)^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3
Can be rewritten as follows:
(a+b)^3 - 3 ab (a + b) - (a^3 + b^3) = 0
If we multiply this by 12 we get:
12 (a+b)^3 - 36 ab (a + b) - 12 (a^3 + b^3) = 0 (1)
Then if we can find two numbers a and b such that
36 ab = -1 (2)
12 (a^3 + b^3) = 1 (3)
Then the identity (1) ...
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