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The solutions to problem 1, 2, and 3 from the week #1 practice quiz of math 412. Problem 1 asks to find a polynomial of smallest degree with real coefficients that has both z1 = 1 + i and z2 = 3 - 2i as roots. Problem 2 requires computing (i - √3)793 and expressing the answer in cartesian form. Problem 3 asks to show that nonzero vectors z1 and z2 are perpendicular if and only if re(z1z2) = 0.
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Problem 1. (8 points) Find a polynomial of smallest degree that has real coefficients such that both z 1 = 1 + i and x 2 = 3 − 2 i are roots.
Problem 2. (6 points) Compute (i −
3)^793 , and express your answer in Cartesian form.
Problem 3. (6 points) Show that the nonzero vectors z 1 and z 2 are perpendicular if and only if Re(z 1 z 2 ) = 0.
