Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 3 - Section 3.4 - Real Zeros of Polynomials - 3.4 Exercises - Page 284: 66

Answer

4 or 2 or 0 real zeros

Work Step by Step

For Descartes' Rule of Signs, the number of sign changes (+ then - or - then +) for P(x) will be the number of possible positive real roots. The number of sign changes for P(-x) will be the number of possible negative real roots. Given $P(x) = x^4 + x^3 + x^2 + x + 12$ For P(x), there are 0 sign changes. Thus, there are 0 possible positive real roots. For $P(-x) = (-x)^4 + (-x)^3 + (-x)^2 + (-x) + 12$ $P(-x) = x^4 - x^3 + x^2 - x + 12$ There are 4 sign changes. Thus there are either 4 or 2 or no real negative real zeros Thus the answer is 4 or 2 or 0 real zeros
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