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
Understanding Polynomial Functions
3:10 minutes
Problem 36a
Textbook Question
Textbook QuestionIn Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^4+6x^3−18x^2; between 2 and 3
Verified Solution
This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mPlay a video:
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 on different signs at the endpoints f(a) and f(b), 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 polynomials.
Recommended video:
6:15
Introduction to Hyperbolas
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) = x^4 + 6x^3 - 18x^2 is a polynomial of degree 4, which is continuous and differentiable everywhere, making it suitable for applying the Intermediate Value Theorem.
Recommended video:
06:04
Introduction to Polynomial Functions
Evaluating Function Values
To apply the Intermediate Value Theorem, it is essential to evaluate the polynomial function at the endpoints of the interval, in this case, f(2) and f(3). By calculating these values, we can determine if they have opposite signs, which indicates the presence of at least one real zero within the interval.
Recommended video:
4:26
Evaluating Composed Functions
Watch next
Master Introduction to Polynomial Functions with a bite sized video explanation from Callie
Start learningRelated Videos
Related Practice