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Math 112
Section 2.3 Polynomial Functions
Polynomial Function
A polynomial function of degree n, where n is a nonnegative integer, is defined by
f ( x)  an x n  an1 x n1      a1 x  a0 , where an , an1 ,...,a1 x, and a0 are real numbers,
called coefficients, with a n  0 . The number a n is called the leading coefficient.
Leading Term:
an x n
an  0
n even
both ends up
a n  0 both ends down
n odd
left down, right up
left up, right down
Determine the leading term, the leading coefficient, the degree of the polynomial, and
the end behavior of the graph:
Example 1: f ( x)  2 x 3  3x 2  5x  4
Example 2: f ( x)   x 4  2 x3  3
1) 2x3
1)  x 4
2) 2
2) -1
3) 3
3) 4
4) left down, right up
4) both ends down
Example 3: f ( x)   x5  3x3  7
Example 4: f ( x)  2 x  3x  2
Finding the zeroes of polynomials:
 Factor
 Substitution rule
Example 5: f ( x)  x 3  2 x 2  5x  6 = (x + 3)(x – 1)(x – 2)
x = -3, 1, 2
Note that f(-3) = 0, f(1) = 0, and f(2) = 0
2
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The multiplicity of a zero is the number of times that zero occurs.
If ( x  c) k , k  1, is a factor, then c is a zero of the function with multiplicity k.
 If k is odd, then the graph crosses the x-axis at (c, 0)
 If k is even, then the graph is tangent to the x-axis at (c, 0) (touches the x-axis
but does not cross)
Find the zeroes, state the multiplicity, and sketch:
Example 6: f ( x)  5( x  2) 3 ( x  1)
x = 2 with multiplicity 3 (crosses at (2,0)); x = -1 with multiplicity 1 (crosses at (1,0))
Example 7: f ( x)  ( x  1) 2 ( x  2) 2
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Things to remember when graphing polynomials:
 The zeroes of an equation are actually the ____________ on the graph.
 The ____________ of a zero tells us if it crosses the x-axis or is tangent to it.
 The __________ of the polynomial will tell us the maximum number of zeros (or
x-intercepts) the polynomial can have.
 If the ___________ of a polynomial is n, then the graph can have at most n-1
turning points.
 The leading term tells us about end behavior
Find the maximum number of real zeros, x-intercepts, and turning points of the
following:
Example 1: f ( x)  5 x  x 8
Example 2: f ( x)  3x 4  x 2  7
Max real zeros: 8
Max x-intercepts: 8
Max turning points: 8-1 = 7
Example 3: Choose the correct graph of h( x)  x( x  4)(x  1)(x  5)
Example 4: Choose the correct graph of f ( x)   x 5  3x 4