PureMath.pdf

# Condition for a purely imaginary root this is easily

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Condition for a purely imaginary root. This is easily found to be C 2 - 4 bBC - 4 b 2 c = 0 . Conditions for a pair of conjugate complex roots. Since the sum and the product of two conjugate complex numbers are both real, b + Bi and c + Ci must both be real, i.e. B = 0, C = 0. Thus the equation (1) can have a pair of conjugate complex roots only if its coefficients are real. The reader should verify this conclusion by means of the explicit expressions of the roots. Moreover, if b 2 = c , the roots will be real even in this case. Hence for a pair of conjugate roots we must have B = 0, C = 0, b 2 < c . 15. The Cubic equation. Consider the cubic equation z 3 + 3 Hz + G = 0 , where G and H are complex numbers, it being given that the equation has ( a ) a real root, ( b ) a purely imaginary root, ( c ) a pair of conjugate roots. If H = λ + μi , G = ρ + σi , we arrive at the following conclusions.

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[III : 46] COMPLEX NUMBERS 110 ( a ) Conditions for a real root. If μ is not zero, then the real root is - σ/ 3 μ , and σ 3 + 27 λμ 2 σ - 27 μ 3 ρ = 0. On the other hand, if μ = 0 then we must also have σ = 0, so that the coefficients of the equation are real. In this case there may be three real roots. ( b ) Conditions for a purely imaginary root. If μ is not zero then the purely imaginary root is ( ρ/ 3 μ ) i , and ρ 3 - 27 λμ 2 ρ - 27 μ 3 σ = 0. If μ = 0 then also ρ = 0, and the root is yi , where y is given by the equation y 3 - 3 λy - σ = 0, which has real coefficients. In this case there may be three purely imaginary roots. ( c ) Conditions for a pair of conjugate complex roots. Let these be x + yi and x - yi . Then since the sum of the three roots is zero the third root must be - 2 x . From the relations between the coefficients and the roots of an equation we deduce y 2 - 3 x 2 = 3 H, 2 x ( x 2 + y 2 ) = G. Hence G and H must both be real. In each case we can either find a root (in which case the equation can be reduced to a quadratic by dividing by a known factor) or we can reduce the solution of the equation to the solution of a cubic equation with real coefficients. 16. The cubic equation x 3 + a 1 x 2 + a 2 x + a 3 = 0, where a 1 = A 1 + A 0 1 i , . . . , has a pair of conjugate complex roots. Prove that the remaining root is - A 0 1 a 3 /A 0 3 , unless A 0 3 = 0. Examine the case in which A 0 3 = 0. 17. Prove that if z 3 +3 Hz + G = 0 has two complex roots then the equation 8 α 3 + 6 αH - G = 0 has one real root which is the real part α of the complex roots of the original equation; and show that α has the same sign as G . 18. An equation of any order with complex coefficients will in general have no real roots nor pairs of conjugate complex roots. How many conditions must be satisfied by the coefficients in order that the equation should have ( a ) a real root, ( b ) a pair of conjugate roots?
[III : 46] COMPLEX NUMBERS 111 19. Coaxal circles. In Fig. 26 , let a , b , z be the arguments of A , B , P .

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