Precalculus: Mathematics for Calculus, 7th Edition

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

Chapter 3 - Review - Concept Check - Page 319: 7

Answer

Remainder Theorem - If the polynomial $P(x)$ is divided by $x-c$, then the remainder is the value $P(c)$ Factor Theorem - $c$ is a zero of $P$ if and only if $x-c$ is a factor of $P(x)$ Rational Zeroes Theorem - If the polynomial $P(x)=a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0$ has integer coeficients (where $a_n\ne0$ and $a_0\ne0$), then every rational zero of $P$ is of the form : $$\frac{p}{q}$$ where $p$ and $q$ are integers and $p$ is a factor of the constant coefficient $a_0$ $q$ is a factor of the leading coefficient $a_n$

Work Step by Step

As also explained in the chapter earlier, the theorems are based on the following ideas : Remainder Theorem - If the polynomial $P(x)$ is divided by $x-c$, then the remainder is the value $P(c)$ Factor Theorem - $c$ is a zero of $P$ if and only if $x-c$ is a factor of $P(x)$ Rational Zeroes Theorem - If the polynomial $P(x)=a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0$ has integer coeficients (where $a_n\ne0$ and $a_0\ne0$), then every rational zero of $P$ is of the form : $$\frac{p}{q}$$ where $p$ and $q$ are integers and $p$ is a factor of the constant coefficient $a_0$ $q$ is a factor of the leading coefficient $a_n$
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