Transcript polynomial function

Section 2-2
Polynomial Functions
End Behavior
Objectives
• I can determine if an equation is a
polynomial in one variable
• I can find the degree of a polynomial
• I can use the Leading Coefficient Test for
end behavior in Limit Notation
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A polynomial function is a function of the form
f ( x )  a n x n  a n 1 x n 1 
 a1 x  a 0 , a n  0
where n is a nonnegative integer and each ai (i = 0, , n)
is a real number. The polynomial function has a leading
coefficient an and degree n.
Examples: Find the leading coefficient and degree of each
polynomial function.
Polynomial Function
f ( x )  2 x 5  3 x 3  5 x  1
Leading Coefficient
Degree
–2
5
f ( x)  x3  6 x 2  x  7
1
3
f ( x )  14
14
0
3
Complex Numbers
Real Numbers
Rationals
Imaginary Numbers
Irrational
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Polynomial Functions/Equations:
A polynomial function in one variable may look like this.
f  x   5x  2 x  4 x  x  3
5
4
3
2
A.The coefficients are complex numbers
(real or imaginary).
B.
Exponents must be a non-negative integer
(zero or positive).
C. The leading coefficient (the coefficient of the
variable with greatest degree) may not be zero.
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Not a polynomial
Polynomials
f  x   3x  x  1
2
f  x   3 x  3x
1/ 2
the exp. is not an integer
f  x   3x
5
3
f  x    4 x  3x 1  4 x
x
the exp. is not non-negative
2
3x
f  x 
1
5
x5
f  x  2
x 4
denominator has a variable factor
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EX:
Determine if each expression is a polynomial in
one variable
f  x   x  2x  4x
4
2
f  s   s  s  5s  6s
4
2
5
3
f  y  3y   9
y
2
YES
3
YES
No
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Practice 1: Given the following equations determine the following:
1. Determine if the equation a polynomial. Why or why not?
2. If the equation is a polynomial what is the degree of each term,
of the polynomial.
A.
f  x   3x 2  x  1
Yes, notice powers on the x are
positive integers and coefficients
are real numbers.
C.
f  x  3 x
No, notice power on the x is the fraction 1/2
xx
1
2
B.
f  x 
3
 4x
x
No, notice power on x
is -1
D.
3
 3x 1
x
3x 2
f  x 
1
5
Yes, notice powers on the x are
positive integers and coefficients
are real numbers.
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Group Exploration
Directions:
Divide into groups of 2
Open your text to page 141.
Read the Exploration exercise
10 minutes!!
Answers on next slides
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
9
Group Exploration
Open the text to page 141.
Read the Exploration exercise instructions.
Use the Leading coefficient test on page 141
a.
b.
f ( x)  x  2 x  x  1
3
2
The leading coefficient is + 1
The degree of the function is 3, that is, f(x) is a cubic.
c.
d.
f ( x)   as x  -
f(x)   as x  
e.
f.
g.
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Group Exploration
Open the text to page 141.
Read the Exploration exercise instructions.
Use the Leading coefficient test on page 141
b.
f ( x)  2 x 5  2 x 2  5 x  1
c.
The leading coefficient is + 2
The degree is 5 and odd.
d.
e.
f.
f ( x)   as x  -
f(x)   as x  
g.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
11
Group Exploration
Open the text to page 141.
Read the Exploration exercise instructions.
Use the Leading coefficient test on page 141
c.
d.
f ( x)  2 x  x  5 x  3
5
2
The leading coefficient is - 2
The degree is 5 and odd.
e.
f.
g.
f ( x)   as x  
f(x)   as x  
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Group Exploration
Open the text to page 141.
Read the Exploration exercise instructions.
Use the Leading coefficient test on page 141
d.
e.
f ( x)   x  5 x  2
3
The leading coefficient is - 1
The degree is 3 and odd.
f.
g.
f ( x)   as x  
f(x)   as x  
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Group Exploration
Open the text to page 141.
Read the Exploration exercise instructions.
Use the Leading coefficient test on page 141
e.
f.
f ( x)  2 x 2  3x  4
The leading coefficient is + 2
The degree is 2 and even.
g.
f(x)  + as x  - 
f(x)  + as x  + 
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Group Exploration
Open the text to page 141.
Read the Exploration exercise instructions.
Use the Leading coefficient test on page 141
f.
g.
f ( x)  x  3x  2 x  1
4
2
The leading coefficient is + 1
The degree is 4 and even.
f(x)  + as x  - 
f(x)  + as x  + 
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Group Exploration
Open the text to page 141.
Read the Exploration exercise instructions.
Use the Leading coefficient test on page 141
g.
f ( x)  x  3x  2
2
The leading coefficient is + 1
The degree is 2 and even.
f(x)  + as x  - 
f(x)  + as x  + 
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
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Leading Coefficient Test
As x grows positively or negatively without bound, the value
f (x) of the polynomial function
f (x) = anxn + an – 1xn – 1 + … + a1x + a0 (an  0)
grows positively or negatively without bound depending upon
the sign of the leading coefficient an and whether the degree n
is odd or even.
y
y
an positive
x
x
n odd
an negative
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
n even
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Example: Describe the right-hand and left-hand behavior
for the graph of f(x) = –2x3 + 5x2 – x + 1.
Degree
Leading Coefficient
3
Odd
-2
Negative
As x   , f ( x )   and as x   , f ( x )  
y
x
f (x) = –2x3 + 5x2 – x + 1
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Closure:
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A real number a is a zero of a function y = f (x)
if and only if f (a) = 0.
Real Zeros of Polynomial Functions
If y = f (x) is a polynomial function and a is a real number then
the following statements are equivalent.
1. (a, 0) is a zero of f.
2. x = a is a solution of the polynomial equation f (x) = 0.
3. (x – a) is a factor of the polynomial f (x).
4. (a, 0) is an x-intercept of the graph of y = f (x).
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Solution or Root
Zero or X-intercept
Factor
x4
x  2
(4, 0)
( x  4)
( 2, 0)
( x  2)
2
x
3
2
( , 0)
3
(3x  2)
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Homework
• WS 3-3
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