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- 1. Chapter 5: Polynomials and Polynomial Functions 5.1: Polynomial Functions
- 2. Definitions <ul><li>A monomial is a real number, a variable, or a product of a real number and one or more variables with whole number exponents. </li></ul><ul><ul><li>Examples: </li></ul></ul><ul><li>The degree of a monomial in one variable is the exponent of the variable. </li></ul>
- 3. Definitions <ul><li>A polynomial is a monomial or a sum of monomials. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><li>The degree of a polynomial in one variable is the greatest degree among its monomial terms. </li></ul><ul><ul><li>Example: </li></ul></ul>
- 4. Definitions <ul><li>A polynomial function is a polynomial of the variable x . </li></ul><ul><ul><li>A polynomial function has distinguishing “behaviors” </li></ul></ul><ul><ul><ul><li>The algebraic form tells us about the graph </li></ul></ul></ul><ul><ul><ul><li>The graph tells us about the algebraic form </li></ul></ul></ul>
- 5. Definitions <ul><li>The standard form of a polynomial function arranges the terms by degree in descending order </li></ul><ul><ul><li>Example: </li></ul></ul>
- 6. Definitions <ul><li>Polynomials are classified by degree and number of terms. </li></ul><ul><ul><li>Polynomials of degrees zero through five have specific names and polynomials with one through three terms also have specific names. </li></ul></ul>Degree Name 0 Constant 1 Linear 2 Quadratic 3 Cubic 4 Quartic 5 Quintic Number of Terms Name 1 Monomial 2 Binomial 3 Trinomial 4+ Polynomial with ___ terms
- 7. Example <ul><li>Write each polynomial in standard form. Then classify it by degree and by number of terms. </li></ul>
- 8. Example <ul><li>Write each polynomial in standard form. Then classify it by degree and by number of terms. </li></ul>
- 9. Polynomial Behavior <ul><li>The degree of a polynomial function </li></ul><ul><ul><li>Affects the shape of its graph </li></ul></ul><ul><ul><li>Determines the number of turning points (places where the graph changes direction) </li></ul></ul><ul><ul><li>Affects the end behavior (the directions of the graph to the far left and to the far right) </li></ul></ul>
- 10. Polynomial Behavior <ul><li>The graph of a polynomial function of degree n has at most n – 1 turning points. </li></ul><ul><ul><li>Odd Degree = even number of turning points </li></ul></ul><ul><ul><li>Even Degree = odd number of turning points </li></ul></ul><ul><li>Think about this: </li></ul><ul><ul><li>If a polynomial has degree 2, how many turning points can it have? </li></ul></ul><ul><ul><li>If a polynomial has degree 3, how many turning points can it have? </li></ul></ul>
- 11. Polynomial Behavior <ul><li>End behavior is determined by the leading term </li></ul>
- 12. Polynomial Behavior Examples
- 13. Example <ul><li>Determine the end behavior of the graph of each polynomial function. </li></ul>
- 14. Example <ul><li>Determine the end behavior of the graph of each polynomial function. </li></ul>
- 15. Increasing and Decreasing <ul><li>Remember: We read from left to right! </li></ul><ul><li>A function is increasing when the y-values increase as the x-values increase </li></ul><ul><li>A function is decreasing when the y-values decrease as the x-values increase </li></ul>
- 16. Example: Identify the parts of the graph that are increasing or decreasing
- 17. Example: Identify the parts of the graph that are increasing or decreasing
- 18. Homework <ul><li>P285 #8 – 31all </li></ul>

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