By Evans, Les.

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**Extra info for Complex Numbers and Vectors (MathWorks for Teachers)**

**Example text**

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.

Fontano was to be their champion. Fior and Fontano each composed 30 equations for the other to solve within 50 days. Fior was overconfident and boasted that Fontano would never be able to solve the equations he had set him. Two hours later Fontano had solved his 30 equations. His success made him famous, but fortune still eluded him. Fontano’s newly found fame reached the ears of Cardano, who was considered to be the most able mathematician in Italy—and he knew it. Cardano was outspoken and highly critical of his colleagues and had a habit of making enemies.

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.