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mat67-2011-hw2 - Homework Assignment#2 Homework due Math 67...

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Homework Assignment #2 Math 67 UC Davis, Fall 2011 Homework due. Tuesday 10/11/11 at discussion section. Reading material. Read Sections 3.2–4.3 in the textbook (the proof of the Fundamental Theorem of Algebra in section 3.1 is optional material). Problems 1. In an earlier lecture we encountered the (non-linear) system of equations x + y = 1 5 x + 2 y = zx 2 x + 8 y = zy which came up in the application of the Google PageRank algorithm. In that example, in order to solve for x, y we had to provide the value of z without explaining how it was obtained. This problem provides that missing explanation. (a) Show how the second and third equations in the system lead to a quadratic equa- tion whose two solutions are the possible values of z , by applying the criterion in “proof-writing exercise” 1 of Chapter 1 in the textbook. (b) For each of the two values of z obtained in part (a), solve the system for x and y , and explain why only one of the solutions makes sense (in terms of the meaning of x and y in the original problem). 2. For each of the following pairs of polynomials f ( x ) and g ( x ), perform polynomial long division with remainder of
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