Math 35
5.3 "Polynomials and Polynomial Functions"
Bibiana Lopez
Riverside Community College
September 2010
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Objectives:
*
*
*
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De…ne and classify polynomials.
Evaluate and graph polynomial functions.
Simplify polynomials by combining like terms.
Add and subtract polynomials.
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De…ne and Classify Polynomials
De…nition
Polynomials
A polynomial is a single term or the sum of terms in which all variables
have whole-number exponents.
Examples:
5x + 3, 4n2
6n
8, p 3 + 3p 2 q + 3pq 2 + q 3 ,
5 2 4
2 rs t
Note: Polynomials can be classi…ed according to their number of terms
and degree.
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De…ne and Classify Polynomials
De…nition
Degree of a Term of a Polynomial The degree of a term of a
polynomial in one variable is the value of the exponent on the variable. If
a polynomial is in more than one variable, the degree of a term is the sum
of the exponents on the variables in that term. The degree of a nonzero
constant is 0 . The constant 0 has no de…ned degree
De…nition
Degree of a Polynomial
The degree of a polynomial is the same as the highest degree of any term
of the polynomial.
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De…ne and Classify Polynomials
Polynomials According to Number of Terms
Name
Number of Terms
Example
Monomial
Binomial
Trinomial
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One term
Two terms
Three terms
5x, 3x 2 , 15 , and 2xy 2 z
x + 3,
x 5 x 3 , and 21 x 3
2
x
2x + 8, and x 8 2x 4 1
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De…ne and Classify Polynomials
Polynomials According to Degree
Name
Degree
Linear
Quadratic
Cubic
First-degree
Second-degree
Third-degree
Example
1
2x
x + 2,
and
3 x
x 2 + 3x 15 and x 2 3x
x 3 2x 2 + x 2 and x 3
x2
Note: A polynomial with four or more terms have no special name.
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De…ne and Classify Polynomials
Example 1: (De…ning and classifying polynomials)
Use the vocabulary of this section to describe each polynomial.
a) x 4 2x 2 + 4
b)
2m12 4m10 n4 + 9m8 n6 + mn9
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Evaluate Polynomial Functions
We have seen that linear (…rst degree) functions are de…ned by
equations of the form f (x ) = mx + b . Examples of linear functions,
f (x ) = 3x + 1, g (x ) = 12 x 1, and h (x ) = 5x .
De…nition
Polynomial Functions
A polynomial function is a function whose equation is de…ned by a
polynomial in one variable.
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Evaluate Polynomial Functions
Polynomial functions can be used to model many real-life situations.
Example 2: (Applications using polynomial functions)
If a toy rocket is shot straight up with an initial velocity of 128 feet per
second, its height, in feet, t seconds after being launched is given by the
function h (t ) = 16t 2 + 128t. Find the height of the rocket 2 seconds
after being launched.
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Graph Polynomial Functions
Three basic polynomial functions:
y
4
2
-4
-2
2
-2
4
x
-4
The Identity Function
Domain:
Range:
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Graph Polynomial Functions
y
4
2
-4
-2
2
-2
4
x
The Squaring function
Domain:
Range:
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Graph Polynomial Functions
y
4
2
-4
-2
2
-2
4
x
-4
The Cubing Function
Domain:
Range:
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Graph Polynomial Functions
Example 3: (Graphing polynomial functions)
Graph f (x ) = x 3 3x 2 9x + 2 and …nd its domain and range.
y
x
f (x ) = x 3
3x 2
9x + 2
8
6
4
2
-8 -6 -4 -2
-2
2
4
6
8
x
-4
-6
-8
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Simplify Polynomials by Combining Like Terms
> Like terms have the same variables with the same exponents.
> When we combine like terms we combine their coe¢ cients and
keep the same variables with the same exponents.
Example 4: (Simplifying and combining like terms)
Simplify each polynomial by combining like terms.
a) 6m4 + 3m4
b) x 19x 2 + 22x 2
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Simplify Polynomials by Combining Like Terms
c) 17s 3 t + 3s 2 t
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6s 3 t
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Simplify Polynomials by Combining Like Terms
d)
7
7
rs + r + 1
8
9
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3
4
rs + r
4
3
2
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Add Polynomials
Adding Polynomials
kTo add polynomials, drop the parentheses and combine their like terms.k
Example 5: (Adding polynomials)
Add the following polynomials.
a) 2a2 3a + 5 + 5a2 + 4a 2
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Add Polynomials
b)
5 5
t
8
5 4
t
6
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+
3 5 3 4
t + t
8
4
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Subtract Polynomials
Subtracting polynomials
To subtract two polynomials, change the signs of the terms of
the polynomial being subtracted, drop the parentheses, and
combine like terms.
Example 6: (Subtracting polynomials)
Subtract the following polynomials.
a) 8x 3 + 2x 2
2x 3 3x 2
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Subtract Polynomials
b)
9m4 + 16m2
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12m4
18m2
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