2.32.3 Polynomial Functions of Higher Degree. Quick Review.
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Transcript of 2.32.3 Polynomial Functions of Higher Degree. Quick Review.
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2.32.32.32.3
Polynomial Functions of Higher Polynomial Functions of Higher DegreeDegree
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Quick Review
2
3 2
2
3
Factor the polynomial into linear factors.
1. 3 11 4
2. 4 10 24
Solve the equation mentally.
3. ( 2) 0
4. 2( 2) ( 1) 0
5. ( 3)( 5) 0
x x
x x x
x x
x x
x x x
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Quick Review Solutions
2
3 2
2
3
Factor the polynomial into linear factors.
1. 3 11 4
2. 4 10 24
Solve the equation mentally.
3. ( 2) 0
4. 2( 2) ( 1) 0
3 1 4
2 2 3 4
0,
5. ( 3)(
2
2, 1
5) 0
x x
x x x
x x
x x
x x x
x x
x x x
x x
x x
0, 3, 5x x x
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What you’ll learn about• How to use transformations to sketch graphs of
Polynomial Functions• How to use the “Leading Coefficient Test” to
determine End Behavior of Polynomial Functions• How to find and use Zeros of Polynomial
Functions• How to use the Intermediate Value Theorem to
locate zeros• … and whyThese topics are important in modeling various
aspects of nature and can be used to provide approximations to more complicated functions.
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Example Graphing Transformations of Monomial
Functions4
Describe how to transform the graph of an appropriate monomial function
( ) into the graph of ( ) ( 2) 5. Sketch ( ) and
compute the -intercept.
n
nf x a x h x x h x
y
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Example Graphing Transformations of Monomial Functions
4
Describe how to transform the graph of an appropriate monomial function
( ) into the graph of ( ) ( 2) 5. Sketch ( ) and
compute the -intercept.
n
nf x a x h x x h x
y
4
4
4
You can obtain the graph of ( ) ( 2) 5 by shifting the graph of
( ) two units to the left and five units up. The -intercept of ( )
is (0) 2 5 11.
h x x
f x x y h x
h
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Cubic Functions
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Quartic Function
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Local Extrema and Zeros of Polynomial
FunctionsA polynomial function of degree n
has at most n – 1 local extrema and at most n zeros.
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Example Applying Polynomial Theory
4 3Describe the end behavior of ( ) 2 3 1 using limits.g x x x x
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Example Applying Polynomial Theory
4 3Describe the end behavior of ( ) 2 3 1 using limits.g x x x x
lim ( )xg x
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Example Finding the Zeros of a Polynomial
Function3 2Find the zeros of ( ) 2 4 6 .f x x x x
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Example Finding the Zeros of a Polynomial
Function
3 2Find the zeros of ( ) 2 4 6 .f x x x x
3 2
Solve ( ) 0
2 4 6 0
2 1 3 0
0, 1, 3
f x
x x x
x x x
x x x
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Example Sketching the Graph of a Factored
Polynomial3 2Sketch the graph of ( ) ( 2) ( 1) .f x x x
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Example Sketching the Graph of a Factored
Polynomial
3 2Sketch the graph of ( ) ( 2) ( 1) .f x x x
The zeros are 2 and 1. The graph crosses the -axis at 2 because
the multiplicity 3 is odd. The graph does not cross the -axis at 1 because
the multiplicity 2 is even.
x x x x
x x
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Intermediate Value Theorem
If a and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other
words, if y0 is between f(a) and f(b), then y0=f(c) for some number c in [a,b].