Transcript Real Zeros of Polynomial Functions Lesson 4.4 Division of Polynomials  Can be done manually   See Example 2, pg 253 Calculator can also do division  Use propFrac(

Real Zeros of
Polynomial Functions
Lesson 4.4
Division of Polynomials
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Can be done manually
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See Example 2, pg 253
Calculator can also do division
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Use propFrac( ) function
Division Algorithm
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For any polynomial f(x) with degree n ≥ 0
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There exists a unique polynomial q(x)
and a number r
Such that f(x) = (x – k) q(x) + r
The degree of q(x) is one less than
the degree of f(x)
The number r is called the remainder
Remainder Theorem
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If a polynomial f(x) is divided by x – k
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The remainder is f(k)
f ( x)  3 x 4  7 x 3  4 x  5
f (2)  11
Factor Theorem
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When a polynomial division results in a
zero remainder
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The divisor is a factor
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f(x) = (x – k) q(x) + 0
This would mean that f(k) = 0
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That is … k is a zero of the function
Completely Factored Form
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When a polynomial is completely factored, we
know all the roots
f ( x )  ( x  a )  ( x  b)  ( x  c )  ( x  d )
roots  a, b, c, d 
Zeros of Odd Multiplicity
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Given f ( x)   x  1  x  3
3
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Zeros of -1 and 3 have odd multiplicity
The graph of f(x) crosses the x-axis
Zeros of Even Multiplicity
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Given f ( x)   x  1  x  3
2
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4
Zeros of -1 and 3 have even multiplicity
The graph of f(x) intersects but does
not cross the x-axis
Try It Out
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Consider the following functions
Predict which will have zeros where
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The graph intersects only
The graph crosses
f ( x)   x  7 
2
 x  3
3
h( x)  x   x  2.5   x  7 
3
4
From Graph to Formula
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If you are given the graph of a polynomial, can
the formula be determined?
Given the graph below:
What are the zeros?
What is a possible
set of factors?
Note the double zero
From Graph to Formula
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Try graphing the results ... does this give the
graph seen above (if y tic-marks are in units
of 5 and the window is -30 < y < 30)
The graph of f(x) = (x - 3)2(x+ 5) will not go
through the point (-3,7.2)
We must determine the coefficient that is the
vertical stretch/compression factor...
f(x) = k * (x - 3)2(x + 5) ... How??
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Use the known point (-3, 7.2)
7.2 = f(-3) Solve for k
Assignment
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Lesson 4.4
Page 296
Exercises 1 – 53 EOO
73 – 93 EOO