College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 3, Polynomial and Rational Functions - Section 3.5 - Complex Zeros and the Fundamental Theorem of Algebra - 3.5 Exercises - Page 330: 44

Answer

$P(x)=4x^{4}+52x^{2}+36$

Work Step by Step

Since $2i, 3i$ are zeros, then so $-2i, -3i$ by the Conjugate Zeros Theorem. This mean that P(x) have the following form: $P(x)=a[(x-2i)(x-(-2i))][(x-3i)(x-(-3i))]$ $P(x)=a(x-2i)(x+2i)(x-3i)(x+3i)$ $P(x)=a(x^{2}-4i^{2})(x^{2}-9i^{2})$ $P(x)=a(x^{4}-9x^{2}i^{2}-4x^{2}i^{2}+36i^{4})$ $P(x)=a(x^{4}+13x^{2}+36)$ To make all all coefficients interger, we set $a=4$ and get $P(x)=4x^{4}+52x^{2}+36$
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