The graph will cross the x-axis at zeros with odd multiplicities. At \(x=3\), the factor is squared, indicating a multiplicity of 2. In this section we will explore the local behavior of polynomials in general. If the graph crosses the x-axis and appears almost We and our partners use cookies to Store and/or access information on a device. Using technology to sketch the graph of [latex]V\left(w\right)[/latex] on this reasonable domain, we get a graph like the one above. 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\newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Recognizing Characteristics of Graphs of Polynomial Functions, Using Factoring to Find Zeros of Polynomial Functions, Identifying Zeros and Their Multiplicities, Understanding the Relationship between Degree and Turning Points, Writing Formulas for Polynomial Functions, https://openstax.org/details/books/precalculus, status 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The graph of a polynomial function will touch the x -axis at zeros with even multiplicities. If the y-intercept isnt on the intersection of the gridlines of the graph, it may not be easy to definitely determine it from the graph. The shortest side is 14 and we are cutting off two squares, so values wmay take on are greater than zero or less than 7. The graph has a zero of 5 with multiplicity 1, a zero of 1 with multiplicity 2, and a zero of 3 with multiplicity 2. (Also, any value \(x=a\) that is a zero of a polynomial function yields a factor of the polynomial, of the form \(x-a)\).(. The bumps represent the spots where the graph turns back on itself and heads Each x-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.

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