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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 36
Chapter 4, Problem 36

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x4+6x3−18x2; between 2 and 3

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First, 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 and the interval: \( f(x) = x^{4} + 6x^{3} - 18x^{2} \), and the interval is between \( 2 \) and \( 3 \).
Evaluate \( f(2) \) by substituting \( x = 2 \) into the function: \( f(2) = 2^{4} + 6(2)^{3} - 18(2)^{2} \).
Evaluate \( f(3) \) by substituting \( x = 3 \) into the function: \( f(3) = 3^{4} + 6(3)^{3} - 18(3)^{2} \).
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 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 when the function changes sign.
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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, making it possible to locate zeros by checking sign changes.
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Evaluating Function Values at Given Points

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

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x4−6x3+14x2−14x+5f(x) = x^4 - 6x^3 + 14x^2 -14x + 5

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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+2x2−4x−8≥0x^3+2x^2−4x−8≥0

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Textbook Question

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x−x2−2

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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

Textbook Question

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

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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)=3x4−2x3−8x+5f(x) = 3x^4 - 2x^3 - 8x + 5