Skip to main content
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 58
Chapter 4, Problem 58

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

Verified step by step guidance
1
Identify the zeros of the polynomial by setting each factor equal to zero: from \((x-2)^2\) we get \(x=2\), and from \((x-5)^2\) we get \(x=5\).
Determine the multiplicity of each zero: both zeros have multiplicity 2, which means the graph will touch the x-axis at these points and turn around (no crossing).
Note the leading term behavior: since the polynomial is \(f(x) = -(x-2)^2 (x-5)^2\), when expanded, the highest degree term will be \(-x^4\), indicating the graph behaves like \(-x^4\) for large \(|x|\) (both ends go down).
Use the sign of the leading coefficient (negative) to confirm that as \(x \to \pm \infty\), \(f(x) \to -\infty\), so the graph falls on both ends.
Combine all this information: the graph touches the x-axis at \(x=2\) and \(x=5\) without crossing, and both ends go down. Use these characteristics to match the correct graph from choices A–F.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7m
Was this helpful?

Key Concepts

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

Polynomial Function and Degree

A polynomial function is an expression involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. The degree of the polynomial is the highest exponent sum in any term, which influences the general shape and end behavior of its graph.
Recommended video:
06:04
Introduction to Polynomial Functions

Zeros and Their Multiplicities

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.
Recommended video:
03:42
Finding Zeros & Their Multiplicity

End Behavior of Polynomial Graphs

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 sign: an even degree with a negative leading coefficient means both ends of the graph point downward.
Recommended video:
06:08
End Behavior of Polynomial Functions
Related Practice
Textbook Question

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

3
views
Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=2x5-x4+2x3-2x2+4x-4; no real zero greater than 1

5
views
Textbook Question

Solve each rational inequality. Give the solution set in interval notation. 8 /(x - 2) ≥ 2

4
views
Textbook Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=f(x)y=f\(\left\)(x\(\right\)). State the domain of ff.

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. 20/(x - 1) ≥ 1

5
views
Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x4+x3-x2+3; no real zero less than -2

3
views