Transcript term - Ctc.edu

Section 5.2
Addition and
Subtraction of
Polynomials
Copyright © 2013 Pearson Education, Inc.
Polynomial
Page 307
A term is a number, a variable, or the product or quotient of a
number and one or more variables raised to powers.
 m5 
1 5
5
9
2
4 x,
m or 
,

7
z
,
6
x
z
,
, and 9.

2
2
3x
 2 
The number in the product is called the numerical coefficient, or
just the coefficient.
8k3
-4p5
8 is the coefficient
–4 is the coefficient
Polynomial
Page 307
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
3x  5,
4m3  5m2 p  8,
Not Polynomials
x 1  3x 2 ,
9  x,
and
and
 5t 2 s3
1
x
Example
Page 307-8
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.
9 y2  7 y  4
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. 7 x 4  2 x 3 y 2  xy  4 y 3
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.
3
8x 
x4
2
The expression is not a polynomial because it contains division by the
polynomial x + 4.
Example
Page 309
State whether each pair of expressions contains like
terms or unlike terms. If they are like terms, then add
them.
The terms have the same variable raised to the
a. 9x3, −2x3
same power, so they are like terms and can be
combined.
9x3 + (−2x3) = (9 + (−2))x3 = 7x3
b. 5mn2, 8m2n
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
Page 309-10
2
2
3
x

4
x

8

4
x
   5 x  3
Add by combining like terms. 
Solution
 3x
2
 4 x  8    4 x 2  5 x  3    3 x 2  4 x  8    4 x 2  5 x  3
 3x 2  4 x 2  4 x  5 x  8  3
 (3  4) x 2  (4  5) x  (8  3)
 7x 2  x  5
Example
Page 310
2
2
2
2
7
x

3
xy

7
y


2
x

xy

2
y
 
.
Simplify. 
Solution
Write the polynomial in a vertical format and then add
each column of like terms.
7 x 2  3 xy  7 y 2
2 x 2  xy  2 y 2
5 x 2  2 xy  5 y 2
Subtraction of Polynomials
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.
Example
3
2
3
2
5
w

3
w

6

5
w

4
w
 8 .
 
Simplify. 
Solution
3
2
3
2
5
w

4
w

8
is

5
w

4
w
 8



The opposite of
  5w3  3w2  6    5w3  4 w2  8 
 (5  5) w3  (3  4) w2  (6  8)
 0 w3  7 w2  2
 7 w2  2
Page 311
Example
Page 311
2
2
10
x

4
x

5

4
x
   2 x  1 .
Simplify. 
Solution
10 x 2  4 x  5
 4 x2  2 x  1
6 x2  6 x  6
Problem 40
Add : (  x 2  x)  (2 x 2  3x  1)
 x  x 
2 x  3x  1
2
2
x2  2x  1
Combine like terms
Problem 66
Subtract : (x 2  3xy  4 y 2 )  ( x 2  xy  4 y 2 )
( x 2  3 xy  4 y 2 )
( x 2  xy  4 y 2 )
4 xy
Add the opposite of the
polynomial being subtracted.
Problem 67
Subtract : (x 2  2 x  3)  (2 x 2  7 x  1)
( x 2  2 x  3)
( 2 x 2  7 x  1 )
- x 2  5x  4
Add the opposite of the
polynomial being subtracted.
Problem 76
Number 76 Area of a Rectangle: Write a polynomial that gives
he area of the rectangle. Calculate its area for x=3 feet.
Area : ( 7  3 x  3 x  x)
3x 2  21x
3(3) 2  21(3)
3  9  21(3)
27  63
90 ft 2
7
x
3x
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: 8, 7 y, x3 , 8 x 2 y9 ,  xy8
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.
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.
Solution
The volume of ONE cube is found by multiplying the
length, width and height.
V  xxx
V  x3
The volume of 3 cubes would be: V  3 x 3
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.
Solution
Volume when x = 4 would be:
V  3x3
V  3(4)3
 192
The volume is 192 square inches.