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
9:46 minutes
Problem 41b
Textbook Question
Textbook QuestionGraph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=2x^3(x^2-4)(x-1)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where 'n' is a non-negative integer and 'a_n' are constants. Understanding the structure of polynomial functions is essential for graphing them accurately.
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Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its simpler polynomial factors. This process is crucial for simplifying expressions and finding the roots of the polynomial, which are the x-values where the function equals zero. Techniques for factoring include identifying common factors, using the difference of squares, and applying the quadratic formula when necessary.
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Graphing Polynomial Functions
Graphing polynomial functions requires understanding their key features, such as intercepts, end behavior, and turning points. The roots of the polynomial, found through factoring, indicate where the graph crosses the x-axis. Additionally, the degree of the polynomial determines the number of turning points and the overall shape of the graph, which is essential for accurately representing the function visually.
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