Assignment 5 Solutions

# Assignment 5 Solutions - Assignment 5 1 What polynomial has...

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Assignment 5 1. What polynomial has roots which are reciprocals of: x 4 + 2 x 3 + 2 x 2 - 10 x - 20. To get reciprocals of roots, simply switch the order of coeﬃcients: - 20 x 4 - 10 x 3 + 2 x 2 + 2 x + 1 2. What polynomial has roots which three times those of 3 x 4 + 2 x 3 - x 2 + 4 x - 5. Simply sub in x 3 for x.(then find the common denominator and factor it out. 3 ( x 3 ) 4 + 2 ( x 3 ) 3 - ( x 3 ) 2 + 4 ( x 3 ) - 5 3 ( x 4 3 4 ) + 2 ( x 3 3 3 ) - ( x 2 3 2 ) + 4 ( x 3 ) - 5 1(3 x 4 ) + 3(2 x 3 ) + 3 2 ( - x 2 ) + 3 3 (4 x ) + 3 4 ( - 5) 81 3 x 4 + 6 x 3 - 9 x 2 + 108 x - 405 81 3 x 4 + 6 x 3 - 9 x 2 + 108 x - 405 3. What polynomial has roots are 1 larger that those of - x 4 - 4 x 3 +14 x 2 + 2 x - 1. Use repeated synthetic division to get the polynomial for x=x-1. - 1 - 1 - 1 - 1 - 1 - 1 - 4 14 2 - 1 1 3 - 17 15 - 1 - 3 17 - 15 14 1 2 - 19 - 1 - 2 19 - 34 1 1 - 1 - 1 20 1 - 1 0 - 1

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So the polynomial we want is - x 4 + 20 x 2 - 34 x + 14. 4. If r is a root of - x 5 + x 4 - 2 x 3 + 2 x 2 + 4 x + 1, then what polynomial has roots 1 3 r . First, find a polynomial with roots of the form 3r, where r is a root of our first polynomial. - ( x 3 ) 5 + ( x 3 ) 4 - 2 ( x 3 ) 3 + 2 ( x 3 ) 2 + 4 ( x 3 ) + 1 - x 5 + 3 * x 4 - 3 2 * 2 x 3 + 3 3 * 2 x 2 + 3 4 * 4 x + 3 5 * 1 273 Which has the same roots as: - x 5 + 3 * x 4 - 3 2 * 2 x 3 + 3 3 * 2 x 2 + 3 4 * 4 x + 3 5 * 1 - x 5 + 3 x 4 - 18 x 3 + 54 x 2 + 324
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