Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 55

Chapter 4, Problem 55

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)²(x-5)²

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Start by identifying the zeros of the polynomial function ƒ(x) = (x-2)^2 (x-5)^2. The zeros are the values of x that make the function equal to zero. Set each factor equal to zero: x - 2 = 0 and x - 5 = 0, so the zeros are x = 2 and x = 5.

Next, determine the multiplicity of each zero. Since both factors are squared, each zero has multiplicity 2. This means the graph will touch the x-axis at these points but will not cross it.

Analyze the end behavior of the polynomial. Since the polynomial is the product of two squared binomials, its degree is 4 (2 + 2), which is even, and the leading coefficient is positive (since the leading terms multiply to a positive x^4 term). Therefore, as x approaches ±∞, ƒ(x) approaches +∞.

Use the information about zeros and end behavior to match the graph: the graph should touch the x-axis at x=2 and x=5 without crossing, and both ends of the graph should rise upwards.

Finally, compare these characteristics with the given graph choices A–F to identify which graph matches the zeros, multiplicities, and end behavior described.

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Key Concepts

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

Polynomial Functions and Their Graphs

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Their graphs are smooth and continuous curves, with shapes influenced by the degree and coefficients of the polynomial. Understanding the general shape helps in matching the function to its graph.

Zeros and Their Multiplicities

The zeros of a polynomial are the values of x that make the function equal to zero. The multiplicity of a zero indicates how many times that root is repeated. Even multiplicities cause the graph to touch the x-axis and turn around, while odd multiplicities cause the graph to cross the axis.

End Behavior of Polynomial Functions

The end behavior describes how the graph behaves as x approaches positive or negative infinity. It depends on the leading term's degree and coefficient. For even-degree polynomials with positive leading coefficients, both ends rise; for even degree with negative coefficients, both ends fall.

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End Behavior of Polynomial Functions

Related Practice

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (2x + 3)/(x - 5) ≤ 0

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Textbook Question

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zeros of -2, 1, and 0; ƒ(-1)=-1

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Textbook Question

For each polynomial function, identify its graph from choices A–F.

$ƒ (x) = - (x - 2)^{2} (x - 5)$

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Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (x + 1)/(x - 5) > 0

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Textbook Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

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Textbook Question

Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = 5x⁴ + 2x³ -x+3; k=2/5

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For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)2(x-5)2

Verified video answer for a similar problem:

Key Concepts

Polynomial Functions and Their Graphs

Zeros and Their Multiplicities

End Behavior of Polynomial Functions

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)²(x-5)²