Skip to main content
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 33
Chapter 4, Problem 33

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x3−x−1; between 1 and 2

Verified step by step guidance
1
Recall the Intermediate Value Theorem (IVT), which states that if a function \( f \) 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 and the interval: \( f(x) = x^3 - x - 1 \), and the interval is \([1, 2]\).
Evaluate \( f(1) \): calculate \( f(1) = 1^3 - 1 - 1 = 1 - 1 - 1 \).
Evaluate \( f(2) \): calculate \( f(2) = 2^3 - 2 - 1 = 8 - 2 - 1 \).
Check the signs of \( f(1) \) and \( f(2) \). If one is negative and the other is positive, then by the IVT, there is at least one real zero of \( f(x) \) between 1 and 2.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 within an interval.
Recommended video:
6:15
Introduction to Hyperbolas

Polynomial Continuity

Polynomials are continuous functions for all real numbers, meaning there are no breaks, jumps, or holes in their graphs. This continuity ensures that the Intermediate Value Theorem can be applied to polynomials on any interval.
Recommended video:
06:04
Introduction to Polynomial Functions

Evaluating Function Values at Interval Endpoints

To apply the Intermediate Value Theorem, you calculate the function values at the endpoints of the interval. If the function values have opposite signs, it indicates the function crosses zero somewhere between those points, confirming the existence of a real root.
Recommended video:
4:26
Evaluating Composed Functions
Related Practice
Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (2−x)2(x−7/2)<0

757
views
Textbook Question

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=(x+7)/(x2+4x−21)

1051
views
Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (1−x)2(x−5/2)<0

380
views
Textbook Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x3−4x2+2; between 0 and 1

1065
views
Textbook Question

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=x3+2x2+5x+4

493
views
Textbook Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers.f(x)=2x4−4x2+1; between -1 and 0

616
views