461exam1sol - Math 461 F Exam 1 - Fri Feb 18 Spring 2011...

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Math 461 F Spring 2011 Exam 1 — Fri Feb 18 Drew Armstrong The exam is worth 25 points. There are 4 problems worth 6 points each, and you get 1 point for writing your name. Problem 1. [6 points] Consider the cubic equation ax 3 + bx 2 + cx + d = 0. (a) What change of variable do you make to get the “depressed” cubic equation? You set x = y - b/ 3 a to get an equation in y with no y 2 term. (This is called the “depressed” cubic.) You can divide the whole equation by a if you want, but this is not so important (I gave full points either way.) (b) Now suppose that the depressed equation has roots - 1 , 0 , 1. In this case, what are the roots of the original? If y = - 1 , 0 , 1 are the roots of the depressed cubic, then the original has roots x = - 1 - b 3 a , x = 0 - b 3 a , x = 1 - b 3 a . (c) Use the information from (b) to completely factor ax 3 + bx 2 + cx + d . Since we know the roots, we can factor the polynomial ax 3 + bx 2 + cx + d to get a ± x + 1 + b 3 a ²± x + b 3 a ²± x - 1 + b 3 a ² . Problem 2.
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This note was uploaded on 01/08/2012 for the course MATH 561 taught by Professor Armstrong during the Spring '11 term at University of Miami.

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461exam1sol - Math 461 F Exam 1 - Fri Feb 18 Spring 2011...

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