Copyright © 2013 Pearson Education, Inc. Section 5.2 Addition and Subtraction of Polynomials.
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Transcript of Copyright © 2013 Pearson Education, Inc. Section 5.2 Addition and Subtraction of Polynomials.
Copyright © 2013 Pearson Education, Inc.
Section 5.2
Addition and Subtraction of Polynomials
Polynomial
55 9 2
2
1 54 , or , 7 , 6 , , and 9.
2 2 3
mx m z x z
x
A term is a number, a variable, or the product or quotient of a number and one or more variables raised to powers.
The number in the product is called the numerical coefficient, or just the coefficient.
8k3 8 is the coefficient-4p5 –4 is the coefficient
Page 307
Polynomial
A polynomial is a term or a finite sum of terms in which all variables have whole number exponents and no variables appear in denominators.
Polynomials
Not Polynomials
3 2 2 33 5, 4 5 8, and 5x m m p t s
1 2 13 , 9 , and x x x
x
Page 307
Example
Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains and its degree. (The degree of monomial is the sum of the exponents of the variables. If the monomial has only one variable, its degree is the exponent of that variable.)
a. The expression is a polynomial with three terms and one variable. The term with the highest degree is 9y2, so the polynomial has degree 2.
b.
The expression is a polynomial with four terms and two variables. The term with the highest degree is 2x3y2, so the polynomial has degree 5.
c.
The expression is not a polynomial because it contains division by the polynomial x + 4.
2 38
4x
x
3234 427 yxyyxx
479 2 yy
Page 307-8
Example
State whether each pair of expressions contains like terms or unlike terms. If they are like terms, then add them.a.9x3, −2x3
b. 5mn2, 8m2n
9x3 + (−2x3) = (9 + (−2))x3 = 7x3
Page 309
The terms have the same variable raised to the same power, so they are like terms and can be combined.
The terms have the same variables, but these variables are not raised to the same power. They are therefore unlike terms and cannot be added.
Example
Add by combining like terms.
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
Page 309-10
Example
Simplify.
SolutionWrite the polynomial in a vertical format and then add each column of like terms.
2 2 2 27 3 7 2 2 .x xy y x xy y
2
2
2
2
7 3 7
2 2
yxy
yx y
x
x
2 25 2 5xyx y
Page 310
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.
Subtraction of PolynomialsPage 310
Example
Simplify.
SolutionThe opposite of
3 2 3 25 3 6 5 4 8 .w 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
Page 311
Example
Simplify.
Solution
2 210 4 5 4 2 1 .x x x x
2
2
10 4 5
4 2 1
x
x
x
x
26 6 6x x
Page 311
Problem 40
132 2
2
xx
xx
1 2 2 xx Combine like terms
)132() :Add 22 xxxx(
Problem 66
Add the opposite of the polynomial being subtracted.
)4()43 :Subtract 2222 yxyxyxy(x
) 4(
)43(22
22
yxyx
yxyx
xy4
Problem 67
Add the opposite of the polynomial being subtracted.
)172()32 :Subtract 22 xxx(x
) 172(
)32( 2
2
xx
xx
45 x- 2 x
Problem 76
)337 :Area xxx(
x21 3x2
Number 76 Area of a Rectangle: Write a polynomial that gives he area of the rectangle. Calculate its area for x=3 feet.
7
x
3x21(3) 3(3)2
2ft 90
6327
21(3) 93
DONE
Objectives
• Monomials and Polynomials
• Addition of Polynomials
• Subtraction of Polynomials
• Evaluating Polynomial Expressions
Monomials and Polynomials
A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers.Examples of monomials: The degree of monomial is the sum of the exponents of the variables. If the monomial has only one variable, its degree is the exponent of that variable.
The number in a monomial is called the coefficient of the monomial.
3 2 9 88, 7 , , 8 , y x x y xy
Example
Write a monomial that represents the total volume of three identical cubes that measure x along each edge. Find the total volume when x = 4 inches.
SolutionThe volume of ONE cube is found by multiplying the length, width and height.
The volume of 3 cubes would be:
3
V x x x
V x
33V x
Example (cont)
Write a monomial that represents the total volume of three identical cubes that measure x along each edge. Find the total volume when x = 4 inches.
SolutionVolume when x = 4 would be:
The volume is 192 square inches.
33V x33(4)
192
V
