\chapter{Polynomial and rational functions}
\section{The Remainder and Factor Theorems}
Answer the questions in your own words.
\begin{enumerate}
\item What does the Remainder Theorem say?
\item We performed synthetic division on the polynomial \(f(x)\) and the final table looks like
| \(2\) |
\(a\) | \(b\) | \(c\) |
\(d\) | \(e\) |
| | | \(u\) | \(v\) | \(w\) | \(x\) |
| | \(y\) | \(z\) |
\(s\) | \(t\) |
\(3\) |
What is \(f(2)\)?
\item Let \(f(x)=3x^9-7x^5+8x^3-9x+8\). We divided \(f(x)\) by \(x+1\) using
synthetic division and our final answer looks like
| \(-1\) |
\(a\) | \(b\) | \(c\) |
\(d\) | \(e\) |
| | | \(u\) | \(v\) | \(w\) | \(x\) |
| | \(y\) | \(z\) |
\(s\) | \(t\) |
\(r\) |
What is \(r\)?
\item Suppose \(f(x)=x^4-x^3+x^2-x+1\), and we want to find \(f(-9)\). Which method is
faster: plug in \(x=-9\), or divide by \(x+9\) with synthetic division?
\item Suppose \(f(x)=3x^{14}-2x^{12}-4x^7-2\), and we want to find the remainder
of the division of \(f(x)\) by \(x-1\). Explain which method you would use, and why.
\item What does the Factor Theorem say?
\item We need to work with the polynomial \(f(x)=3x^8-7x^5+3x^2+8x+k\), and we
can change \(k\) to
be any number we like. Which number should \(k\) be to make sure that \(x+1\) is a factor of \(f(x)\)?
\end{enumerate}