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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 56
Chapter 4, Problem 56

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


Six graphs labeled A to F show different polynomial curves with x-intercepts at 2 and 5 for the function f(x) = (x-2)(x-5).

Verified step by step guidance
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Start by identifying the zeros of the polynomial function ƒ(x) = (x-2)(x-5). 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.
Solve each equation to find the zeros: x = 2 and x = 5. These are the x-intercepts of the graph, meaning the graph crosses the x-axis at these points.
Determine the degree and leading coefficient of the polynomial. Since ƒ(x) is the product of two linear factors, it is a quadratic polynomial of degree 2. When expanded, the leading term will be \(x^2\), which has a positive coefficient.
Use the leading coefficient and degree to determine the end behavior of the graph. Because the leading coefficient is positive and the degree is even, the graph opens upwards on both ends.
Combine this information: the graph crosses the x-axis at x=2 and x=5, opens upwards, and is a parabola. Use these characteristics to match the function to the correct graph among choices A–F.

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Key Concepts

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

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. Understanding the general shape and behavior of polynomial functions helps in predicting their graphs.
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Introduction to Polynomial Functions

Roots or Zeros of a Polynomial

The roots or zeros of a polynomial are the values of x that make the function equal to zero. For ƒ(x) = (x-2)(x-5), the roots are x = 2 and x = 5, which correspond to the x-intercepts on the graph.
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Imaginary Roots with the Square Root Property

Graphing Polynomial Functions

Graphing involves plotting key points such as roots and determining the end behavior based on the leading term. For a quadratic like ƒ(x) = (x-2)(x-5), the graph is a parabola opening upwards, crossing the x-axis at the roots.
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Graphing Polynomial Functions
Related Practice
Textbook Question

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zero of -3 having multiplicity 3; ƒ(3)=36

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

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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

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