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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 5.1 Adding and Subtracting Polynomials Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
Section 5.1
Adding andSubtractingPolynomials
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 2
Objective #1 Understand the vocabulary used to
describe polynomials.
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Polynomials
A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents.
Consider the polynomial: 6523 34 xxx
This polynomial contains four terms. It is customary to write the terms in order of descending powers of the variable. This is the standard form of a polynomial. Here are two other polynomials which are written in standard form.
638
82754
23
xx
xxx
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The Degree of axn • If a ≠ 0, the degree of axn is n. The degree of a nonzero constant
term is 0. The constant 0 has no defined degree.
8275 23 xxx
Degree 3 Degree 2 Degree 1
Degree of nonzero constant: 0
Polynomials
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• The degree of a polynomial is the degree of its highest order term.
• Example:
Degree 3 Polynomial:
Degree 4 Polynomial:
3 2
4
5 7 2 8
8 3 6
x x x
x x
Degree of a Polynomial
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• Monomial: A polynomial with one term.
• Binomial: A polynomial with two terms.
• Trinomial: A polynomial with three terms.
Example:
This is a 4th degree trinomial.
638 4 xx
Special Polynomials
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Objective #1: Example
1. Identify the polynomial as a monomial, a binomial, or a trinomial. Give the degree of the polynomial.
5 37 3 8x x
5 37 3 8x x is a trinomial of degree 5.
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Objective #1: Example
1. Identify the polynomial as a monomial, a binomial, or a trinomial. Give the degree of the polynomial.
5 37 3 8x x
5 37 3 8x x is a trinomial of degree 5.
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Objective #2 Add polynomials.
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Polynomials
Adding PolynomialsPolynomials are added by removing the parentheses that surround each polynomial (if any) and then combining like terms.
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• Polynomials are added by combining like terms.
• Like terms are terms containing exactly the same variables to the same powers.
• Example:2222 10)64(64 xxxx
These like terms both contain x to the second power.
Add the coefficients and keep the same variable
factor.
Adding Polynomials
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Adding Polynomials
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Add: . 1771119131167 2323 xxxxxx
1771119131167 2323 xxxxxx
1771119131167 2323 xxxxxx Remove parentheses
4 4 5 12
1713711116197
23
2233
xxx
xxxxxx Rearrange terms so that like terms are adjacent
Combine like terms
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Objective #2: Example
2a. Add the polynomials:
3 2 3 2( 11 7 11 5) (16 3 3 15)x x x x x x
3 2 3 2
3 2 3 2
3 3 2 2
3 2
( 11 7 11 5) (16 3 3 15)
11 7 11 5 16 3 3 15
11 16 7 3 11 3 5 15
5 4 8 20
x x x x x x
x x x x x x
x x x x x x
x x x
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Objective #2: Example
2a. Add the polynomials:
3 2 3 2( 11 7 11 5) (16 3 3 15)x x x x x x
3 2 3 2
3 2 3 2
3 3 2 2
3 2
( 11 7 11 5) (16 3 3 15)
11 7 11 5 16 3 3 15
11 16 7 3 11 3 5 15
5 4 8 20
x x x x x x
x x x x x x
x x x x x x
x x x
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Objective #2: Example
2b. Add 3 211 7 11 5x x x and 3 216 3 3 15x x x using a vertical format.
3 2
3 2
3 2
11 7 11 5
16 3 3 15
5 4 8 20
x x x
x x x
x x x
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Objective #2: Example
2b. Add 3 211 7 11 5x x x and 3 216 3 3 15x x x using a vertical format.
3 2
3 2
3 2
11 7 11 5
16 3 3 15
5 4 8 20
x x x
x x x
x x x
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Objective #3 Subtract polynomials.
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Polynomials
Subtracting PolynomialsTo subtract two polynomials, add the first polynomial and the opposite of the polynomial being subtracted.
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Objective #3: Example
3a. Subtract 3 23 8 5 6x x x from 3 210 5 7 2.x x x
3 2 3 2
3 2 3 2
3 3 2 2
3 2
(10 5 7 2) (3 8 5 6)
10 5 7 2 3 8 5 6
10 3 5 8 7 5 2 6
7 3 12 8
x x x x x x
x x x x x x
x x x x x x
x x x
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Objective #3: Example
3a. Subtract 3 23 8 5 6x x x from 3 210 5 7 2.x x x
3 2 3 2
3 2 3 2
3 3 2 2
3 2
(10 5 7 2) (3 8 5 6)
10 5 7 2 3 8 5 6
10 3 5 8 7 5 2 6
7 3 12 8
x x x x x x
x x x x x x
x x x x x x
x x x
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Objective #3: Example
3b. Use the method of subtracting in columns to find 3 2 3(8 10 14 2) (5 3 6).y y y y y
3 2
3
8 10 14 2
5 3 6
y y y
y y
To subtract, add the opposite of the polynomial being subtracted.
3 2
3
3 2
8 10 14 2
5 3 6
3 10 11 8
y y y
y y
y y y
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Objective #3: Example
3b. Use the method of subtracting in columns to find 3 2 3(8 10 14 2) (5 3 6).y y y y y
3 2
3
8 10 14 2
5 3 6
y y y
y y
To subtract, add the opposite of the polynomial being subtracted.
3 2
3
3 2
8 10 14 2
5 3 6
3 10 11 8
y y y
y y
y y y
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Objective #4 Graph equations defined by polynomials
of degree 2.
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Graphs of Polynomial Functions
Graphs of equations defined by polynomials of degree 2, such as have a mirror like quality. We can obtain their graphs, shaped like bowls or inverted bowls, using the point-plotting method for graphing an equation in two variables.
2 4,y x
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Objective #4: Example
4. Graph the equation 2 1y x . Make a table of values using integers from 3 to 3. Table of values.
2
2
2
2
2
2
2
2
1 ( , )
3 ( 3) 1 8 3,8
2 ( 2) 1 3 2,3
1 ( 1) 1 0 1,0
0 (0) 1 1 0, 1
1 (1) 1 0 1,0
2 (2) 1 3 2,3
3 (3) 1 8 3,8
x y x x y
y
y
y
y
y
y
y
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Objective #4: Example
4. Graph the equation 2 1y x . Make a table of values using integers from 3 to 3. Table of values.
2
2
2
2
2
2
2
2
1 ( , )
3 ( 3) 1 8 3,8
2 ( 2) 1 3 2,3
1 ( 1) 1 0 1,0
0 (0) 1 1 0, 1
1 (1) 1 0 1,0
2 (2) 1 3 2,3
3 (3) 1 8 3,8
x y x x y
y
y
y
y
y
y
y
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Objective #4: Example
CONTINUED
Plot the points and connect them with a smooth curve.
