Polynomial functions
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Polynomial Functions. Lesson 9.2. Polynomials. Definition: The sum of one or more power function Each power is a non negative integer. Polynomials. General formula a 0 , a 1 , … ,a n are constant coefficients n is the degree of the polynomial

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Polynomial Functions

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Polynomial functions

Polynomial Functions

Lesson 9.2


Polynomials

Polynomials

  • Definition:

    • The sum of one or more power function

    • Each power is a non negative integer


Polynomials1

Polynomials

  • General formula

    • a0, a1, … ,an are constant coefficients

    • n is the degree of the polynomial

    • Standard form is for descending powers of x

    • anxn is said to be the “leading term”


Polynomial properties

Polynomial Properties

  • Consider what happens when x gets very large negative or positive

    • Called “end behavior”

    • Also “long-run” behavior

  • Basically the leading term anxn takes over

  • Comparef(x) = x3 with g(x) = x3 + x2

    • Look at tables

    • Use standard zoom, then zoom out


Polynomial properties1

Polynomial Properties

  • Compare tables for low, high values


Polynomial properties2

The leading term x3 takes over

For 0 < x < 500the graphs are essentially the same

Polynomial Properties

  • Compare graphs ( -10 < x < 10)


Zeros of polynomials

Zeros of Polynomials

  • We seek values of x for which p(x) = 0

  • Consider

    • What is the end behavior?

    • What is q(0) = ?

    • How does this tell us that we can expect at least two roots?


Methods for finding zeros

Methods for Finding Zeros

  • Graph and ask for x-axis intercepts

  • Use solve(y1(x)=0,x)

  • Use zeros(y1(x))

  • When complex roots exist, use cSolve() or cZeros()


Practice

Practice

  • Giveny = (x + 4)(2x – 3)(5 – x)

    • What is the degree?

    • How many terms does it have?

    • What is the long run behavior?

  • f(x) = x3 +x + 1 is invertible (has an inverse)

    • How can you tell?

    • Find f(0.5) and f -1(0.5)


Assignment

Assignment

  • Lesson 9.2

  • Page 400

  • Exercises 1 – 29 odd


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