Polynomial Functions and Models

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Section 3.1

• Polynomial Functions and Models

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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
  a0 is
• the constant coefficient or constant term. The
• number an, the coefficient of the highest power,
  is
• the leading coefficient, and the term anxn is the
  
• leading term.

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Example of a Polynomial Function
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Graphs of Polynomial Functions and Nonpolynomial
Functions
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Graphs of Polynomials

• Graphs are smooth curves
• 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

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End Behavior of Polynomials

• End Behavior- a description of what happens as x
  becomes large in the positive and negative
  direction.
• End Behavior is determined by
• Term with the highest power of x
• Sign of this terms coefficient

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Even- and Odd-Degree Functions
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The Leading-Term Test
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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

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X-Intercepts (Real Zeros)

• A polynomial function of degree n will have at
  most n x-intercepts (real zeros).

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Number of Turning Points (relative maxima/minima)

• The number of relative maxima/minima of the
  graph of a polynomial function of degree n is at
  most n 1.
• ex. f(x) x4 3x3 2x2 1
• Determine number of relative maxima/minima
• n 1 4 1 3

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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
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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)

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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.
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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.  
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True or False?

• 1.) The function
  must
• have 1 real zero.
• 2.) The function
  has no real zeros.
• 3.) An odd degree polynomial function must have
  at least 1 real zero.
• 4.) An even degree polynomial function must have
  at least 1 real zero.

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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.

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The Intermediate Value Theorem

• Consider a polynomial function P(x) with the
  points (a, P(a)) and (b, P(b)) on the function.
• For any P(x) with real coefficients, suppose
  that for a ? b, P(a) and P(b) are of opposite
  signs. Then the function has a real zero between
  a and b.

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The Intermediate Value Theorem

• In other words, if one point is above the
  x-axis and the other point is below the x-axis,
  then because P(x) is continuous and will have to
  cross the x-axis to connect the two points, P(x)
  must have a zero somewhere between a and b.