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  • 1. Chapter 5: Polynomials andPolynomial Functions5.1: Polynomial Functions
  • 2. DefinitionsA monomial is a real number, a variable, or a product of a real number and one or more variables with whole number exponents. 2 3 ◦ Examples: 5, x, 3wy , 4 xThe degree of a monomial in one variable is the exponent of the variable.
  • 3. DefinitionsA polynomial is a monomial or a sum of monomials. ◦ Example: 3 xy + 2 x − 5 2The degree of a polynomial in one variable is the greatest degree among its monomial terms. ◦ Example: −4 x − x + 7 2
  • 4. DefinitionsA polynomial function is a polynomial of the variable x. ◦ A polynomial function has distinguishing “behaviors”  The algebraic form tells us about the graph  The graph tells us about the algebraic form
  • 5. DefinitionsThe standard form of a polynomial function arranges the terms by degree in descending order ◦ Example: P( x) = 4 x + 3x + 5 x − 2 3 2
  • 6. DefinitionsPolynomials are classified by degree and number of terms. ◦ Polynomials of degrees zero through five have specific names and polynomials with one through three terms also have specific names.Degree Name Number of Name0 Constant Terms1 Linear 1 Monomial2 Quadratic 2 Binomial3 Cubic 3 Trinomial4 Quartic 4+ Polynomial with ___ terms5 Quintic
  • 7. ExampleWrite each polynomial in standard form.Then classify it by degree and by numberof terms.3x + 9 x + 5 2
  • 8. ExampleWrite each polynomial in standard form.Then classify it by degree and by numberof terms.3 − 4 x + 2 x +10 5 2
  • 9. Polynomial BehaviorThe degree of a polynomial function ◦ Affects the shape of its graph ◦ Determines the number of turning points (places where the graph changes direction) ◦ Affects the end behavior (the directions of the graph to the far left and to the far right)
  • 10. Polynomial BehaviorThe graph of a polynomial function of degree n has at most n – 1 turning points. ◦ Odd Degree = even number of turning points ◦ Even Degree = odd number of turning pointsThink about this: ◦ If a polynomial has degree 2, how many turning points can it have? ◦ If a polynomial has degree 3, how many turning points can it have?
  • 11. Polynomial BehaviorEnd behavior is determined by the n leading term ax
  • 12. Polynomial Behavior Examplesy = 4x + 6x − x 4 3 y = x3 y = − x2 + 2x y = − x3 + 2 x
  • 13. ExampleDetermine the end behavior of the graphof each polynomial function.y = 4 x − 3x 3
  • 14. ExampleDetermine the end behavior of the graphof each polynomial function.y = −2 x + 8 x − 8 x + 2 4 3 2
  • 15. Increasing and DecreasingRemember: We read from left to right!A function is increasing when the y- values increase as the x-values increaseA function is decreasing when the y- values decrease as the x-values increase
  • 16. Example: Identify the parts of thegraph that are increasing ordecreasing
  • 17. Example: Identify the parts of thegraph that are increasing ordecreasing
  • 18. HomeworkP285 #8 – 31all