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conjugate pair theorem

Conjugate Pair Theorem

An assertion about the complexzeros of any polynomial which has real numbers as coefficients.

Theorem:

If a polynomial

p(x) = anxn + an1xn1 + ··· + a2x2 + a1x + a0

has real coefficients, then any complex zeros occur in conjugate pairs. That is, if a + bi is a zero then so is a – bi and vice-versa.

Example:

2 – 3i is a zero of p(x) = x3 – 3x2 + 9x + 13 as shown here:

p(2 – 3i) = (2 – 3i)3 – 3(2 – 3i)2 + 9(2 – 3i) + 13
              = (–46 – 9i) – 3(–5 – 12i) + (18 – 27i) + 13
              = –46 – 9i + 15 + 36i + 18 – 27i + 13
              = 0.

By the conjugate pair theorem, 2 + 3i is also a zero of p(x).

p(2 + 3i) = (2 + 3i)3 – 3(2 + 3i)2 + 9(2 + 3i) + 13
              = (–46 + 9i) – 3(–5 + 12i) + (18 + 27i) + 13
              = –46 + 9i + 15 – 36i + 18 + 27i + 13
              = 0.

See also

Complex conjugate, polynomial facts

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