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Factor Theorem

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The following result, which is an immediate consequence of the Remainder Theorem:


Let f(x) be a polynomial. Then xh is a factor of f(x) if and only if f(h)=0.

The theorem is valuable for finding factors of polynomials. For example, to factorize 2x 3+3x 2−12x−20, look first for possible factors xh, where h is an integer. Here h must divide 20. Try possible values for h, and calculate f(h). It is found that f(−2)=−16+12 +24−20=0, and so x+2 is a factor. Now divide the polynomial by this factor to obtain a quadratic which it may be possible to factorize further.

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