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Polynomials of Higher Degree2-2
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Polynomials and Their Graphs
Polynomials will always be continuous
Polynomials will always have smooth turns.
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End Behavior(leading coefficient test
Odd Power Positive leading coefficient
Negative leading coefficient
Even Degree Positive leading coefficient
Negative leading coefficient
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Zeros
If f(x) is a polynomial, the following statements are equivalent X = a is a zero of the function X = a is a solution to the equation f(x)=0 (x-a) is a factor of the polynomial f(x) a is an x-intercept of the graph of f(x)
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Finding zeros
Find all real zeros of
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Analyzing a polynomial
Find all real zeros of
Use your calculator to find all relative Extrema (min/max)
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Finding the Zeros of a Polynomial
Graph the function on your calculator, then use it to find the roots.
10
8
6
4
2
-2
-4
-6
-8
-10
-12
-15 -10 -5 5 10 15 20 25
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Repeated Root
If a root or factor is repeated it behaves differently on the graph.
If is a root and k is even it touches the axis and does not cross
If is a root and k is odd it flattens out and crosses the axis
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Repeated Roots
K is even ex.
3
2.5
2
1.5
1
0.5
-0.5
-1
-1.5
-2
-2.5
-3
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
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Repeated Roots
K is odd ex. 3
2.5
2
1.5
1
0.5
-0.5
-1
-1.5
-2
-2.5
-3
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
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Pg 108 # 17-33, 35, 41, 43