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Unread 01-02-2005, 22:18
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Re: Factoring 3rd degree polynomial

Another quick thing I decided to point out that helps.

The rational root theorem states that the only roots of a polynomial equation with integer coefficients must be a rational number where the numerator is a factor of the last coefficient and the denominator is a factor of the first coefficient.

For example, all roots of
x^3 + 1 must be in the set {+- 1}.
5x^3 + 1 must be in the set{+- 1, +- 1/5}.
15x^3 + 4x^2 + 4 must be in the set {+-1, +-2, +-4, +-2/3, +-2/5, +-2/15, +-4/3, +-4/5, +-4/15}.

Note that this tells you only what CANNOT be a root. Numbers that satisfy this condition must still be checked for roothood* by synthetic division.
In addition, also note that "last coefficient" and "first coefficient" refer to coefficients in these positions when the polynomial is organized with the higher power of the variable first.

--Hope this random knowledge helps you one day.

*roothood - the property of being a root (fictitious word)
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