4.2 Polynomial Functions and Models

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4.2Polynomial Functions and Models
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A polynomial function is a function of the form
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f
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3
4
5
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-

Polynomial.
(b)
Not a polynomial.
(c)
Not a polynomial.
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If f is a polynomial function and r is a real
number for which f(r)0, then r is called a
(real) zero of f, or root of f. If r is a (real)
zero of f, then
(a) r is an x-intercept of the graph of f.
(b) (x-r) is a factor of f.
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Use the above to conclude that x -1 and x 4
are the real roots (zeroes) of f.
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1 is a zero of multiplicity 2. -3 is a zero of
multiplicity 1. -5 is a zero of multiplicity 5.
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If r is a Zero or Even Multiplicity
If r is a Zero or Odd Multiplicity
.
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Theorem
If f is a polynomial function of degree n, then f
has at most n-1 turning points.
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Theorem
For large values of x, either positive or
negative, the graph of the polynomial

resembles the graph of the power function.


















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For the polynomial
(a) Find the x- and y-intercepts of the graph of
f.
(b) Determine whether the graph crosses or
touches the x-axis at each x-intercept.
(c) Find the power function that the graph of f
resembles for large values of x.
(d) Determine the maximum number of turning
points on the graph of f.
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For the polynomial
(e) Use the x-intercepts and test numbers to find
the intervals on which the graph of f is above
the x-axis and the intervals on which the graph
is below the x-axis.
(f) Put all the information together, and connect
the points with a smooth, continuous curve to
obtain the graph of f.
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(a) The x-intercepts are -4, -1, and 5.
y-intercept


(b) -4 is a zero of multiplicity 1. (crosses)
-1 is a zero of multiplicity 2. (touches) 5
is a zero of multiplicity 1. (crosses)
(d) At most 3 turning points.
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Test number -5
f (-5) 160
Graph of f Above x-axis
Point on graph (-5, 160)
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-4 lt x lt-1
Test number -2
f (-2) -14
Graph of f Below x-axis
Point on graph (-2, -14)
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-1 lt x lt 5
Test number 0
f (0) -20
Graph of f Below x-axis
Point on graph (0, -20)
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Test number 6
f (6) 490
Graph of f Above x-axis
Point on graph (6, 490)
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(6, 490)
(-1, 0)
(-5, 160)
(0, -20)
(5, 0)
(-2, -14)
(-4, 0)