Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 10 - Systems of Equations and Inequalities - Section 10.5 Partial Fraction Decomposition - 10.5 Assess Your Understanding - Page 787: 4



Work Step by Step

The fundamental theorem of algebra states that a polynomial of degree $n$ with real coefficients has exactly $n$ complex roots, counting the multiplicity of each root. a) For a real root $k$ of the polynomial $A(x)$, then $(x-k)$ is a linear factor of $A(x)$. b) For a complex root, $k=a+bi$, then its conjugate $\overline{k}=a-bi$ is also a root of P. Both $(x-k)$ and $(x- \overline{k})$ are factors of $A(x)$. $(x-a-bi)(x-a+bi)=(x-a)^{2}-(bi)^{2}\\ =x^{2}-2ax+a^{2}+b^{2}$ This shows a quadratic irreducible factor of $A(x)$, thus, the given statement is true.
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