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Factor Theorem

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Factor Theorem - Lesson Summary

Factor Theorem

Let p(x) be a polynomial of degree n>1 and 'a' be a real number. If p(a) = 0, then (x – a) is a factor of p(x). Conversely p(a) = 0, if (x – a) is a factor of p(x).

If p(x) is a polynomial of degree n>1, then p(x) = (x - a) . q(x) + p(a).

If p(a) = 0, then p(x) = (x – a) . q(x). This proves that (x – a) is a factor of p(x).

  • (x + a) is a factor of a polynomial p(x), if and only if p(– a) = 0
  • (ax – b) is a factor of a polynomial p(x), if and only if p(b/a) = 0
  • (ax + b) is a factor of a polynomial p(x), if and only if p(– b/a) = 0
  • (x – a)(x – b) is a factor of a polynomial p(x), if and only if p(a) = 0 and p(b) = 0.

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