Polynomials

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POLYNOMIALS
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Polynomials

• A polynomial is a function of the form

where the
are real numbers and n is a nonnegative integer.
The domain of a polynomial function is the set
of real numbers
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The Degree of Polynomial Functions
The Degree, of a polynomial function in one
variable is the largest power of x Example
Below is a polynomial of degree 2
See Page 183 for a summary of the properties of
polynomials of degree less than or equal to two
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Properties of Polynomial Functions

• The graph of a polynomial function is a smooth
  and continuous curve
• A smooth curve is one that contains no Sharp
  corners or cusps

A polynomial function is continuous if its graph
has no breaks, gaps or holes
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Power Functions

• A power function if degree n, is a function of
  the form

where a is a real number, and n gt 0 is an
integer
Examples (degree 4),
(degree 7) , (degree 1)
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Graphs of even power functions
The polynomial function is even if n 2 is
even. The functions graphed above are even. Note
as n gets larger the graph becomes flatter near
the origin, between (-1, 1), but increases when
x gt 1 and when x lt -1. As x gets bigger and
bigger, the graph increases rapidly.
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Properties of an even function

• The domain of an even function is the set of
  real numbers
• Even functions are symmetric with the
• y-axis
• The graph of an even function contains the points
  (0, 0) (1,1) (-1, 1)

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Graphs of odd power functions
The polynomial function is odd if n 3 is
odd. The functions graphed above are odd. Note
as n gets larger the graph becomes flatter near
the origin, -1 lt x lt1 but increases when x gt 1
or decreases when x lt -1 . As x gets bigger
and bigger, the graph increases for values of x
greater than 1 and decreased rapidly for values
of x less than or equal to -1.
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Properties of an odd function
The domain of an odd function is the set of real
numbers Odd functions are symmetric with the
origin The graph of an odd function contains the
points (0, 0) (1,1) (-1,-1)
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Graphs of Odd functions
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Graphs of Even functions
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Zeros of a polynomial function

• A real number r is a real zero of the polynomial
  f (x) if f (r) 0
• If r is a zero of the polynomial, then r is an x
  intercept.
• If r is a zero of the polynomial f (x) then
• f (x) (x r) p (x), where p (x) is a
  polynomial

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The intercepts of a polynomial

• If r is an x intercept of a polynomial x, then
• f( r ) 0
• If r is an x intercept then either
• 1. The graph crosses the x axis at r or
• 2. The graph touches the x axis at r

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