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Warmup
Write a polynomial with zeros ­4, multiplicity 1
­1, multiplicity 1
3, multiplicity 2
ply it out :)
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Write the polynomial. Multiply it out.
37) Zeros: ­1, 1, 3; degree 3
43) Zeros: ­1, multiplicity 1; 3, multiplicity 2; degree 3
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Who cares about multiplicity?
Graph f(x) = x2(x ­ 2). What are the x­intercepts? What are their multiplicities?
If r is a zero of even multiplicity
Sign of f(x)
Graph
If r is a zero of odd multiplicity
Sign of f(x)
Graph
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Turning points
Use your calculator. How many turning points do you see in...
y = x3
y = x3 ­ x
y = x3 +3x2 + 4
y = x4
y = x4 ­ 2x3
y = x4 ­ 2x2
How does the number of turning points compare with the degree?
If f is a polynomial function of degree n, then f has at most n ­ 1 turning points.
If the graph of a polynomial function f has n ­ 1 turning points, the degree of f is at least n.
If a polynomial function is of degree 5, how many turning points can it have?
If a polynomial function has 7 turning points, what can its degree be?
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Is it a polynomial?
If so, list the real zeros and the least degree possible.
If not, why not?
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Graph these. Then zoom out (change the window) so you can see what is happening for large values of x, either positive or negative (meaning, as x heads towards positive and negative infinity). (Feel free to use Zoom 3 Enter) Notice anything?
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End Behavior
For large values of x, regardless of positive or negative, the graph of the polynomial
resembles the graph of the power function
The end behavior is what happens to the graph for very large values of x.
End behavior of n ≥ 2 even; an > 0
n ≥ 2 even; an < 0
n ≥ 3 odd; an > 0
n ≥ 3 odd; an < 0
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f(x) = ­4(x2 + 1)(x ­ 2)3
Find each real zero and its multiplicity.
Does the graph touch or cross the x­axis at each zero?
What is the maximum possible number of turning points?
What is the end behavior?
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