Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
4. Polynomial Functions
Graphing Polynomial Functions
Problem 39Blitzer - 8th Edition
Textbook Question
In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x^3−10x+9; between -3 and -2
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Identify the function: .
Evaluate the function at the first integer: Calculate .
Evaluate the function at the second integer: Calculate .
Check the sign of and : If one is positive and the other is negative, then there is a real zero between -3 and -2.
Apply the Intermediate Value Theorem: Since is continuous, and and have opposite signs, there must be at least one real zero between -3 and -2.
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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 on different signs at the endpoints, then there exists at least one c in (a, b) such that f(c) = 0. This theorem is crucial for proving the existence of real zeros in polynomial functions.
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Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) = 3x^3 - 10x + 9 is a cubic polynomial, which is continuous and differentiable everywhere, making it suitable for applying the Intermediate Value Theorem.
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Introduction to Polynomial Functions
Sign Change
A sign change occurs when the value of a function changes from positive to negative or vice versa. To apply the Intermediate Value Theorem, we evaluate the polynomial at the endpoints of the interval, -3 and -2, to check for a sign change, indicating the presence of at least one real zero within that interval.
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Related Practice
Textbook Question
Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=2x^4
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