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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 40
Chapter 4, Problem 40

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x3−8x2+x+2; between 2 and 3

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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 interval given: between 2 and 3, so \(a = 2\) and \(b = 3\).
Evaluate the polynomial \(f(x) = 3x^3 - 8x^2 + x + 2\) at the endpoints: calculate \(f(2)\) and \(f(3)\).
Check the signs of \(f(2)\) and \(f(3)\). If one is positive and the other is negative, then by the IVT, there is at least one real zero between 2 and 3.
Conclude that since \(f\) is a polynomial (and thus continuous everywhere) and the function values at 2 and 3 have opposite signs, there must be a real zero of \(f(x)\) between 2 and 3.

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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 within an interval.
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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.
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Evaluating Function Values at Interval Endpoints

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

For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros. f(x)=x3+3x2−4f(x) = x^3 + 3x^2 - 4

Textbook Question

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x4+10x3+5x2+x+1; f(−2/3)

Textbook Question

Use Descartes' Rule of Signs to explain why 2x4+6x2+8=02x^4 + 6x^2 + 8 = 0 has no real roots.

Textbook Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x3/(3x2+1)

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Textbook Question

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=3x2−12x−1

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. x3+x2+4x+4>0x^3+x^2+4x+4>0

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