12. [Maximum mark: 20) (a) P(z) is a polynomial of degree four with real coefficients. Given that 2e and Zie's are roots

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12 Maximum Mark 20 A P Z Is A Polynomial Of Degree Four With Real Coefficients Given That 2e And Zie S Are Roots 1 (43.33 KiB) Viewed 55 times

12. [Maximum mark: 20) (a) P(z) is a polynomial of degree four with real coefficients. Given that 2e and Zie's are roots of the equation. Express P(z) as a product of 2 quadratic factors, (z+ az + b)(z? + cz + d) such that a,b,c,d are real numbers. (b) The following diagram shows the complex plane with origin o, and the points P and Q represents the complex numbers zz and Z1Zz respectively. OP is a third of the length of OQ and the angle POQ is. Im WIN P) Re [Diagram not drawn to scale] Find zz. giving your answer in the form a +ib, a, b ER. (c)(0) Show that for complex numbers, z and w, [4] zw* +zºw = 2Re(zw"). [2] (ii) Hence or otherwise, use proof by contradiction to prove the triangle inequality for complex numbers, that is, Iz + wl <lz+lwl. [71