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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 52
Chapter 4, Problem 52

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=x4-4x3-x+3; 0.5 and 1

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Recall the Intermediate Value Theorem (IVT), which states that if a function \( f(x) \) is continuous on a closed interval \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one \( c \) in \((a, b)\) such that \( f(c) = 0 \).
Identify the function \( f(x) = x^4 - 4x^3 - x + 3 \) and the interval \([0.5, 1]\). Since \( f(x) \) is a polynomial, it is continuous everywhere, including on this interval.
Evaluate \( f(0.5) \) by substituting \( x = 0.5 \) into the function: \( f(0.5) = (0.5)^4 - 4(0.5)^3 - 0.5 + 3 \).
Evaluate \( f(1) \) by substituting \( x = 1 \) into the function: \( f(1) = (1)^4 - 4(1)^3 - 1 + 3 \).
Check the signs of \( f(0.5) \) and \( f(1) \). If one is positive and the other is negative, then by the IVT, there is at least one real zero of \( f(x) \) between 0.5 and 1.

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Key Concepts

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

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes values f(a) and f(b) at each end, then it must take any value between f(a) and f(b) at some point within the interval. This theorem is used to prove the existence of roots when the function changes sign.
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Polynomial Continuity

Polynomial functions are continuous everywhere on the real number line, meaning there are no breaks, jumps, or holes in their graphs. This continuity ensures that the Intermediate Value Theorem can be applied to any interval when analyzing polynomial functions.
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Evaluating Function Values at Interval Endpoints

To apply the Intermediate Value Theorem, you must calculate the function's values at the given interval endpoints. If the function values have opposite signs, it indicates the function crosses zero within the interval, confirming the existence of a real root.
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Related Practice
Textbook Question

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x2+x−2)5(x−1+3)2ƒ(x)=(x^2+x-2)^5(x-1+\(\sqrt\)3)^2

Textbook Question

Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note following Example 3.)

Textbook Question

Graph each polynomial function. ƒ(x)=-2x4+7x3-4x2-4x

Textbook Question

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(2x2−7x+3)3(x−2−5)ƒ(x)=(2x^2-7x+3)^3(x-2-\(\sqrt\)5)

Textbook Question

Work each problem. Which function has a graph that does not have a horizontal asymptote?

A. ƒ(x)=(2x-7)/(x+3)

B. ƒ(x)=3x/(x2-9)

C. ƒ(x)=(x2-9)/(x+3)

D. ƒ(x)=(x+5)/(x+2)(x-3)

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) = x3 +7x2 + 10x; k=0