By Evans, Les.
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Extra info for Complex Numbers and Vectors (MathWorks for Teachers)
When the real quadratic factors of a polynomial are reduced to linear factors, these factors will be conjugate pairs. This property of the factorisation of polynomials with real coefficients is called the Conjugate Root Theorem. CO N J U G AT E RO OT T H EO R E M The non-real roots of a polynomial expression that has only real coefficients occur as conjugate pairs. Some students believe that the conjugate root theorem applies to all complex polynomials. 1 with examples in which the conjugate root theorem does not apply.
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R if and only if z = z 43 MATHSWORKS FOR TEACHERS Complex Numbers and Vectors 1+i We can now return to the division 2 + 3i . We can simplify this problem by multiplying the numerator and denominator by the conjugate of the denominator. 1 + i 2 - 3i 2 + 3 + 2i - 3i 2 + 3i # 2 - 3 i = 4+9 5-i = 13 i 5 = 13 - 13 In general terms: z1 x1 + iy1 z2 = x2 + iy2 x + iy x - iy = x 1 + iy1 # x2 - iy2 2 2 2 2 = x1 x2 + y1 y2 + i^x2 y1 - x1 y2h x22 + y22 This process is analogous to the rationalisation of a real surd expression.