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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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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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