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Intro to Polynomials
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Intro to Polynomials

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  • 1. Polynomial Functions and Models
    Module 12
  • 2. Polynomial Functions
    A polynomial of degree n is a function of the form
    P(x) = anxn + an-1xn-1 + ... + a1x + a0
    Where an 0. The numbers a0, a1, a2, . . . , an are
    called the coefficients of the polynomial.
    The a0is the constant coefficientorconstant term.
    The number an, the coefficient of the highest
    power, is the leading coefficient, and the term anxn is
    the leading term.
  • 3. Graphs of Polynomial Functions and Nonpolynomial Functions
  • 4. Graphs of Polynomials
    Graphs smooth curve
    Degree greater than 2
    ex. f(x) = x3
    These graphs will not have the following:
    Break or hole
    Corner or cusp
    Graphs are lines
    Degree 0 or 1
    ex. f(x) = 3 or f(x) = x – 5
    Graphs are parabolas
    Degree 2
    ex. f(x) = x2 + 4x + 8
  • 5. Even- and Odd-Degree Functions
  • 6. The Leading-Term Test
  • 7. Finding Zeros of a Polynomial
    Zero- another way of saying solution
    Zeros of Polynomials
    Solutions
    Place where graph crosses the x-axis
    (x-intercepts)
    Zeros of the function
    Place where f(x) = 0
  • 8. Using the Graphing Calculator to Determine Zeros
    Graph the following polynomial function and determine the zeros.
    Before graphing, determine the end behavior and the number
    of relative maxima/minima.
    In factored form:
    P(x) = (x + 2)(x – 1)(x – 3)²
  • 9. MultiplicityIf (x-c)k, k 1, is a factor of a polynomial function P(x) and:
    K is even
    The graph is tangent to the x-axis at (c, 0)
    K is odd
    The graph crosses the x-axis at (c, 0)
  • 10. Multiplicity
    y = (x + 2)²(x − 1)³
     Answer.  
     −2 is a root of multiplicity 2,
    and 1 is a root of multiplicity 3.  
    These are the 5 roots:
    −2,  −2,  1,  1,  1.
  • 11. Multiplicity
    y = x³(x + 2)4(x − 3)5
    Answer.   
    0 is a root of multiplicity 3,
    -2 is a root of multiplicity 4,
    and 3 is a root of multiplicity 5.  
  • 12. To Graph a Polynomial
    Use the leading term to determine the end behavior.
    Find all its real zeros (x-intercepts).
    Set y = 0.
    Use the x-intercepts to divide the graph into intervals and choose a test point in each interval to graph.
    Find the y-intercept. Set x = 0.
    Use any additional information (i.e. turning points or multiplicity) to graph the function.