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Polynomials

Polynomials. Lecture 3C Pre AP & GT Precalculus. Agenda. Degrees Humps End Behavior Zeroes. Degrees. **A, B, C, D, E, F, G are all constant coefficients. Other Names. Zeroes…. (definition) A zero of a polynomial is any value x such that

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Polynomials

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  1. Polynomials Lecture 3CPre AP & GT Precalculus

  2. Agenda • Degrees • Humps • End Behavior • Zeroes

  3. Degrees **A, B, C, D, E, F, G are all constant coefficients

  4. Other Names

  5. Zeroes… • (definition) A zero of a polynomial is any value x such that • Find zeroes is a very important and practical problem • Quadratic Equation

  6. A teacher’s motto… • Engage… • Enlighten… • Empower

  7. Zeroes… • Reflection: Is it possible to find a Cubic Equation? • Yes, first found by Scipione del Ferro • HOW: • Depressing the cubic… • Substitute: • Substitute some more: • Quadratic Equation

  8. The Posers Cardano Tartaglia

  9. Cubic EquationSummary Method (Lots of substitution) • Notice the imaginary numbers showing up in x2 and x3 • Notice x1, apparently a cubic always has at least one real root

  10. Quartic Equation • Found by Cardano’s student Ferrari • Published in Cardano’sbook Ars Magna • How messy is the quartic equation?

  11. Quintic Equation Niels Abel Paolo Ruffini Solution to the general Quintic is Impossible!

  12. Smooth or Not Smooth? Smooth Not Smooth This cannot be a polynomial • A polynomial is always smooth

  13. My Humps • If p(x) is an degree n polynomial, then it will has (at most) n-1 humps • If a polynomial has n-1 humps then it is of the nth-degree (at least) • What degree? • 5, 6, 7 ….

  14. My humps, my humpsName the polynomial: Cubic: 2 humps Quintic: 4 humps

  15. Odd Degree End Behavior Positive Leading Coefficient Negative Leading Coefficient Increases Left & Right Decreases Left & Right

  16. Even Degree End Behavior Positive Leading Coefficient Negative Leading Coefficient Decreases LeftIncreases Right Increases LeftDecreases Right

  17. What could it look like?Humps and End Behavior… Septic? • How many humps? • One less so 6 • End behavior? • Increases both left and right

  18. What is a Bump? • Mathematically, a bump is a change of concavity (officially known as a point of inflection) • A bump is equivalent to 2 humps • Example: y=x3 • How many humps do you expect? • 2 but instead you only get1 bump

  19. A note about humps…. • What degree? • Minimum: 16th degree (a bump is two humps: think y=x3)

  20. Finding Zeroes on your calculator • Graph Function…Adjust window so you can see all humps • Draw graph on your paper • Estimate zeroes and the find exact values via Calc (2nd trace), option 2 • Zeroes will be covered in greater depth next week

  21. Challenge Problem (Due Tomorrow) • If where a, b, and c, are constants and then what is • 2 Points for the first correct solution to me (after school) • 1 Point for each correct solution afterwards • 1 Point for presenting it to your class (first taker each period)

  22. Homework • Pg 148 V1,3,5,7 • Pg 148 #1-8(A) • Pg 149 #16,20 • Pg 149 #28-40(4th) • Pg 152 99-101

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