Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 5.1 Polynomial Functions
Mar 26, 2015
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Section 5.1
Polynomial Functions
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Objectives
• Monomials and Polynomials
• Addition and Subtraction of Polynomials
• Polynomial Functions
• Evaluating Polynomials
• Operations on Functions
• Applications and Models
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Monomials and Polynomials
A term is a number, a variable, or a product of numbers and variables raised to powers.
Examples of terms:
If the variables in a term have only nonnegative integer exponents, the term is called a monomial.
Examples of monomials:3 2 9 88, 7 , , 8 , y x x y xy
5 1/2 2 1 812, , , 8 , 8z x x y x y
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Example
Determine whether the expression is a monomial. a. b. c. d.
Solutiona. b.
c. d.
4 79x y 3ab 48w x 3
y
4 79x y 3ab
48w x3
y
monomialnot a monomialnegative exponent
not a monomialsum (+) of two monomials
not a monomialnegative exponent y-1 since 3/y = 3y-1
Division by a variable
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Monomials
The degree of a monomial equals the sum of the exponents of the variables. A constant term has degree 0, unless the term is 0 (which as an undefined degree). The numeric constant in a monomial is called its coefficient. The table shows the degree and coefficient of several monomials.
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Polynomials
A polynomial is either a monomial or a sum of monomials.Polynomials containing one variable are called polynomials of one variable. The leading coefficient of a polynomial of one variable is the coefficient of the monomial with highest degree.The degree of a polynomial equals the degree of the monomial with the highest degree.
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Addition and Subtraction
We can add like terms.If two terms contain the same variables raised to the same power, we call them like terms.
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Example
Simplify each expression by combining like terms.a. b.
Solutiona.
b.
2 2 47 3x x x 2 2 2 28 3 2ab a ab a
2 2 47 3x x x 2 4( )7 3 x x
2 44x x
2 2 2 28 3 2ab a ab a 2 2 2 28 13 2ab a ab a
2 2( )1 3( 28 )ab a
2 27ab a
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Example
Simplify the expression.
Solution
2 23 4 8 4 5 3x x x x
2 28 3443 5x x x x
2 2 4 8 34 53x x x x
2 23 4 8 4 5 3x x x x
2 4( ) (3 4 )3) (85x x
2 57x x
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Example
Find the sum.
SolutionPolynomials can be added vertically by placing like terms in the same columns and then adding each column.
2 2 2 27 3 7 2 2x xy y xy y x
2
2
2
2
7 3 7
2 2
yxy
yx y
x
x
2 25 2 5xyx y
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Subtracting Polynomials
To subtract two polynomials we add the first polynomial to the opposite of the second polynomial. To find the opposite of a polynomial, we negate each term.
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Example
Simplify.SolutionThe opposite of
3 2 3 25 3 6 5 4 8w w w w
3 2 3 25 4 8 is 5 4 8w w w w
3 2 3 25 3 6 5 4 8w w w w
3 2(5 5) (3 4) ( 6 8)w w
3 20 7 2w w 27 2w
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Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Polynomial Functions
The following expressions are examples of polynomials of one variable.
As a result, we say that the following are symbolic representations of polynomial functions of one variable.
2 3, , and 51 515 3x xx x
32( ) 3, , and ( 5 1 ( )) 1 5 5f g x x xx xx hx
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Example
Determine whether f(x) represents a polynomial function. If possible, identify the type of polynomial function and its degree. a.
b.
c.
3( ) 6 2 7f x x x
3.5( ) 4f x x
4( )
5f x
x
cubic polynomial, of degree 3
not a polynomial function because the variable is negative
not a polynomial
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Example
A graph of is shown. Evaluate f(1) graphically and check your result symbolically.
3( ) 5f x x x
To calculate f(–1) graphically find –1 on the x-axis and move down until the graph of f is reached. Then move horizontally to the y-axis.
f(1) = –4 3( 1) 5( ( )1 1)f
5 1
4
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Example
Evaluate f(x) at the given value of x.
Solution
3 2( ) 4 3 7, 2f x x x x
3 2( ) 4 3 7, 2f x x x x
3 2( 2) 4( 2) 3( 2) 7f
( 2) 4( 8) 3(4) 7f
( 2) 32 12 7f
( 2) 27f
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Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Let f(x) = 3x2 + 1 and g(x) = 6 – x2. Find each sum or difference. Solutiona.
b.
2 ( ) 3(
4
1 1) 1
f 2( ) 61 ( )
5
1g
2( ) 3( ) 1
3(4) 1
1
2
3
2
f 2( ) 6 (
6 4
2
2 )2g
( )( 2) ( 2) ( 2)
13 2
11
f g f g
( )(1) (1) (1)
4 5
9
f g f g
a. ( )(1) b. ( )( 2)f g f g
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Example
Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. a. What is the initial heart rate when the athlete stops exercising?
2( ) 1.875 30 200P t t t
2( ) 1.875 30 200 P t t t
When the athlete stops exercising, the heart rate is 200 beats per minute.
2(0) 1.875(0) 30(0) 200
(0) 200
P
P
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Example (cont)
Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. b. What is the heart rate after 8 minutes?
2( ) 1.875 30 200P t t t
2( ) 1.875 30 200 P t t t2(8) 1.875(8) 30(8) 200
(0) 80 beats per minute.
P
P
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Example (cont)
Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. c. A graph of P is shown. Interpret this graph.
The heart rate does not drop at a constant rate; rather, it drops rapidly at first and then gradually begins to level off.
2( ) 1.875 30 200P t t t