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(
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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
Can be done manually
See Example 2, pg 253
Calculator can also do division
Use propFrac( ) function
Division Algorithm
For any polynomial f(x) with degree n ≥ 0
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
If a polynomial f(x) is divided by x – k
The remainder is f(k)
f ( x) 3 x 4 7 x 3 4 x 5
f (2) 11
Factor Theorem
When a polynomial division results in a
zero remainder
The divisor is a factor
f(x) = (x – k) q(x) + 0
This would mean that f(k) = 0
That is … k is a zero of the function
Completely Factored Form
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
Given f ( x) x 1 x 3
3
Zeros of -1 and 3 have odd multiplicity
The graph of f(x) crosses the x-axis
Zeros of Even Multiplicity
Given f ( x) x 1 x 3
2
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
Consider the following functions
Predict which will have zeros where
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
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
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??
Use the known point (-3, 7.2)
7.2 = f(-3) Solve for k
Assignment
Lesson 4.4
Page 296
Exercises 1 – 53 EOO
73 – 93 EOO