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Chapter 5: Polynomials and Polynomial Functions 5.1: Polynomial Functions
Definitions <ul><li>A  monomial  is a real number, a variable, or a product of a real number and one or more variables wit...
Definitions <ul><li>A  polynomial  is a monomial or a sum of monomials. </li></ul><ul><ul><li>Example:  </li></ul></ul><ul...
Definitions <ul><li>A  polynomial function  is a polynomial of the variable  x . </li></ul><ul><ul><li>A polynomial functi...
Definitions <ul><li>The  standard form of a polynomial function  arranges the terms by degree in descending order </li></u...
Definitions <ul><li>Polynomials are  classified  by degree and number of terms. </li></ul><ul><ul><li>Polynomials of degre...
Example <ul><li>Write each polynomial in standard form. Then classify it by degree and by number of terms. </li></ul>
Example <ul><li>Write each polynomial in standard form. Then classify it by degree and by number of terms. </li></ul>
Polynomial Behavior <ul><li>The degree of a polynomial function  </li></ul><ul><ul><li>Affects the shape of its graph </li...
Polynomial Behavior <ul><li>The graph of a polynomial function of degree  n  has  at most   n   – 1  turning points. </li>...
Polynomial Behavior <ul><li>End behavior is determined by the leading term  </li></ul>
Polynomial Behavior Examples
Example <ul><li>Determine the end behavior of the graph of each polynomial function. </li></ul>
Example <ul><li>Determine the end behavior of the graph of each polynomial function. </li></ul>
Increasing and Decreasing <ul><li>Remember: We read from left to right! </li></ul><ul><li>A function is  increasing  when ...
Example: Identify the parts of the graph that are increasing or decreasing
Example: Identify the parts of the graph that are increasing or decreasing
Homework <ul><li>P285 #8 – 31all </li></ul>
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  1. 1. Chapter 5: Polynomials and Polynomial Functions 5.1: Polynomial Functions
  2. 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. 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. 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. 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. 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. 7. Example <ul><li>Write each polynomial in standard form. Then classify it by degree and by number of terms. </li></ul>
  8. 8. Example <ul><li>Write each polynomial in standard form. Then classify it by degree and by number of terms. </li></ul>
  9. 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. 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. 11. Polynomial Behavior <ul><li>End behavior is determined by the leading term </li></ul>
  12. 12. Polynomial Behavior Examples
  13. 13. Example <ul><li>Determine the end behavior of the graph of each polynomial function. </li></ul>
  14. 14. Example <ul><li>Determine the end behavior of the graph of each polynomial function. </li></ul>
  15. 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. 16. Example: Identify the parts of the graph that are increasing or decreasing
  17. 17. Example: Identify the parts of the graph that are increasing or decreasing
  18. 18. Homework <ul><li>P285 #8 – 31all </li></ul>
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