Textbook Question
Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x3-37x2+50x+60 between 7 and 8
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Chapter 4, Problem 32a
Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x3-37x2+50x+60 between 2 and 3
Verified step by step guidance1
First, identify the polynomial function given: \(f(x) = 4x^3 - 37x^2 + 50x + 60\).
Evaluate the function at the endpoints of the interval given, which are \(x=2\) and \(x=3\). Calculate \(f(2)\) and \(f(3)\) by substituting these values into the polynomial.
Check the signs of \(f(2)\) and \(f(3)\). If \(f(2)\) and \(f(3)\) have opposite signs (one positive and one negative), then by the Intermediate Value Theorem, there must be at least one real zero between 2 and 3.
Explain the Intermediate Value Theorem: it states that if a continuous function changes sign over an interval, it must cross zero at some point within that interval.
Conclude that since polynomial functions are continuous everywhere, and if the sign change is confirmed, the function \(f(x)\) has at least one real zero 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.
Polynomial Functions and Their Zeros
A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The zeros of a polynomial are the values of x for which the function equals zero. Understanding how to find or estimate these zeros is essential for analyzing the behavior of the function.
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Finding Zeros & Their Multiplicity
Intermediate Value Theorem
The Intermediate Value Theorem states that if a continuous function changes sign over an interval [a, b], then it must have at least one root in that interval. This theorem is useful for proving the existence of a zero between two points by evaluating the function's values at those points.
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6:15Introduction to Hyperbolas
Evaluating Polynomial Functions at Specific Points
To apply the Intermediate Value Theorem, you need to calculate the polynomial's value at given points. By substituting x-values into the polynomial and finding the sign of the output, you can determine if a zero lies between those points based on a sign change.
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02:44Maximum Turning Points of a Polynomial Function
Related Practice
Textbook Question
Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-x(x+1)(x-1)
Textbook Question
Solve each polynomial inequality. Give the solution set in interval notation. (x + 3)3(2x - 1)(x + 4) ≥ 0
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Textbook Question
Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x^3-37x^2+50x+60 Find the zero in part (b) to three decimal places.
Textbook Question
Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 Find the zero in part (b) to three decimal places.
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Textbook Question
Factor into linear factors given that k is a zero. (multiplicity )