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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 57
Chapter 4, Problem 57

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=2x4−3x3−7x2−8x+6

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Start by finding the zeros of the polynomial function f(x)=2x4−3x3−7x2−8x+6 by setting f(x) = 0 and solving for x.
Use the Rational Root Theorem to list possible rational zeros. These are of the form \(\pm\) \(\frac{p}{q}\), where p divides the constant term 6 and q divides the leading coefficient 2. So possible roots are \(\pm\) 1, \(\pm\) 2, \(\pm\) 3, \(\pm\) 6, \(\pm\) \(\frac{1}{2}\), \(\pm\) \(\frac{3}{2}\).
Test each possible rational root by substituting into the polynomial or using synthetic division to check if it yields zero. When a root is found, factor it out from the polynomial.
After factoring out the first root, reduce the polynomial degree and repeat the process to find other zeros. Continue factoring until the polynomial is completely factored into linear and/or irreducible quadratic factors.
Once all zeros are found, analyze the multiplicity of each zero to understand the behavior of the graph at those points. Use this information along with the leading coefficient and degree to sketch the complete graph without a graphing utility.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Finding Zeros of Polynomial Functions

Zeros of a polynomial are the values of x for which the function equals zero. To find them, one can use factoring, synthetic division, or the Rational Root Theorem to test possible roots. Identifying all zeros is essential for understanding the behavior and shape of the graph.
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Finding Zeros & Their Multiplicity

End Behavior of Polynomial Functions

The end behavior describes how the function behaves as x approaches positive or negative infinity. It depends on the leading term's degree and coefficient. For example, a positive leading coefficient with an even degree means the graph rises on both ends, guiding the sketching of the complete graph.
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End Behavior of Polynomial Functions

Sketching Polynomial Graphs Without Technology

Drawing a polynomial graph by hand involves plotting zeros, determining multiplicities, analyzing end behavior, and finding key points like local maxima and minima. Understanding these features helps create an accurate, complete graph without relying on graphing utilities.
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Graphing Polynomial Functions
Related Practice
Textbook Question

Follow the seven steps to graph each rational function. f(x)=4x/(x−2)

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Textbook Question

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In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)

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Textbook Question

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Textbook Question

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