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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
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Lesson 13-1
Polynomials
Lesson 13-2
Adding Polynomials
Lesson 13-3
Subtracting Polynomials
Lesson 13-4
Multiplying a Polynomial by a Monomial
Lesson 13-5
Linear and Nonlinear Functions
Lesson 13-6
Graphing Quadratic and Cubic Functions
Example 1 Classify Polynomials
Example 2 Degree of a Monomial
Example 3 Degree of a Polynomial
Example 4 Degree of a Real-World Polynomial
Determine whether
is a polynomial. If it is,
classify it as a monomial, binomial, or trinomial.
Answer: The expression is not a polynomial because
has a variable in the denominator.
Determine whether
is a polynomial. If it is,
classify it as a monomial, binomial, or trinomial.
Answer: This is a polynomial because it is the difference
of two monomials. There are two terms, so it is
a binomial.
Determine whether each expression is a polynomial. If
it is, classify it as a monomial, binomial, or trinomial.
a.
Answer: yes; trinomial
b.
Answer: not a polynomial
Find the degree of
.
Answer: The variable w has degree 4, so the degree
of –10w4 is 4.
Find the degree of
has degree 3,
.
has degree 7, and z has degree 1.
Answer: The degree of
Find the degree of each monomial.
a.
Answer: 3
b.
Answer: 8
Find the degree of
term
degree
4
7
0
.
Answer: The greatest degree is 7. So, the degree of the
polynomial is 7.
Find the degree of
term
.
degree
4
7
Answer: The greatest degree is 7. So, the degree of the
polynomial is 7.
Find the degree of each polynomial.
a.
Answer: 6
b.
Answer: 5
Area The formula for the surface area (A) of a cube
is
, where s is the side length. Find the degree
of the polynomial.
Answer:
Area The formula for the surface area S of a cylinder
with height h and radius r is
.
Find the degree of the polynomial.
Answer: 2
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Example 1 Add Polynomials
Example 2 Use Polynomials to Solve a Problem
Find
.
Method 1 Add vertically.
Align like terms.
Add.
Method 2 Add horizontally.
Associative and
Commutative
Properties
Answer: The sum is 10w + 1.
Find
.
Method 1 Add vertically.
Align like terms.
Add.
Method 2 Add horizontally.
Write the expression.
Group like terms.
Simplify.
Answer: The sum is
Find
.
Write the expression.
Simplify.
Answer: The sum is
Find
.
Leave a space because
there is no other term like xy.
Answer: The sum is
.
Find each sum.
a.
Answer:
b.
Answer:
c.
Answer:
d.
Answer:
Geometry The length of a rectangle is
units and the width is 8x – 1 units.
Find the perimeter.
Formula for the
perimeter of a rectangle
Replace with
and w with
Distributive Property
Group like terms.
Simplify.
Answer: The perimeter is
Find the length of the rectangle if
Write the expression.
Replace x with –3.
Simplify.
Answer: The length of the rectangle is 16 units.
Geometry The length of a rectangle is
units and the width is 6w – 3 units.
a. Find the perimeter.
Answer:
b. Find the length if
Answer: 39 units
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Example 1 Subtract Polynomials
Example 2 Subtract Using the Additive Inverse
Example 3 Subtract Polynomials to Solve a Problem
Find
.
Align like terms.
Subtract.
Answer: The difference is
.
Find
.
Align like terms.
Subtract.
Answer: The difference is
.
Find each difference.
a.
Answer:
b.
Answer:
Find
.
To subtract (3x + 9),
add (–3x – 9).
Group the like terms.
Simplify.
Answer: The difference is x–17.
Find
.
The additive inverse of
Align the like terms and add the additive inverse.
Answer:
Find each difference.
a.
Answer: 10c – 7.
b.
Answer:
Geometry The length of a rectangle is
units. The width is
units. How much longer is
the length than the width?
difference in measurement
Substitution
Add additive inverse.
Group like terms.
Simplify.
Answer: The length is
than the width.
units longer
Profit The ABC Company’s costs are given by
where x = the number of items produced.
The revenue is given by 5x. Find the profit, which is
the difference between the revenue and the cost.
Answer:
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Example 1 Products of a Monomial and a Polynomial
Example 2 Product of a Monomial and a Polynomial
Example 3 Use a Polynomial to Solve a Problem
Find
.
Distributive Property
Simplify.
Answer: – 24x – 16
Find
.
Distributive Property
Simplify.
Answer:
Find each product.
a. 3(–5m – 2)
Answer: –15m – 6
b. (4p – 8)(–3p)
Answer:
Find
Distributive Property
Simplify.
Answer:
Find
Answer:
Fences The length of a dog run is 4 feet more than
three times its width. The perimeter of the dog run is
56 feet. What are the dimensions of the dog run?
Explore
You know the perimeter of the dog run. You
want to find the dimensions of the dog run.
Plan
Let w represent the width of the dog run.
Then 3w + 4 represents the length. Write an
equation.
Perimeter equals twice
the sum of the
length and width.
P
=
2
Solve
Write the equation.
Replace P with 56 and
Combine like terms.
Distributive Property
Subtract 8 from each side.
Divide each side by 8.
Answer: The width of the dog run is 6 feet,
and the length is
Examine Check the reasonableness of the results.
The answer checks.
Garden The length of a garden is four more than
twice its width. The perimeter of the garden is 44 feet.
What are the dimensions of the garden?
Answer: 6 feet by 16 feet
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Example 1 Identify Functions Using Graphs
Example 2 Identify Functions Using Equations
Example 3 Identify Functions Using Tables
Example 4 Describe a Linear Function
Determine whether the graph represents a linear or
nonlinear function.
Answer: The graph is a straight
line, so it represents a
linear function.
Determine whether the graph represents a linear or
nonlinear function.
Answer: The graph is a curve,
not a straight line, so it
represents a nonlinear
function.
Determine whether each graph represents a linear or
nonlinear function.
a.
b.
Answer: nonlinear
Answer: linear
Determine whether
nonlinear function.
represents a linear or
Answer: This equation represents a linear function
because it is written in the form
Determine whether
nonlinear function.
represents a linear or
Answer: This equation is nonlinear because x is raised
to the second power and the equation cannot be
written in the form
Determine whether each equation represents a linear
or nonlinear function.
a.
Answer: nonlinear
b.
Answer: linear
Determine whether the table represents a linear or
nonlinear function.
+2
+2
+2
x
2
4
6
8
y
25
17
9
1
–8
As x increases by 2,
y decreases by 8.
So, this is a linear function.
–8
–8
Answer: linear
Determine whether the table represents a linear or
nonlinear function.
+3
+3
+3
x
5
8
11
14
y
2
4
8
16
+2
+4
+8
As x increases by 3,
y increases by a greater
amount each time. So, this is
a nonlinear function.
Answer: nonlinear
Determine whether each table represents a linear or
nonlinear function.
a.
b.
x
3
5
7
9
y
10
11
13
16
Answer: nonlinear
x
10
9
8
7
y
4
7
10
13
Answer: linear
Multiple-Choice Test Item Which rule describes a
linear function?
A
B
C
D
Read the Test Item
A rule describes a relationship between variables. A rule
that can be written in the form
describes a
relationship that is linear.
Solve the Test Item
This is a nonlinear function because x is in the
denominator and the equation cannot be written
in the form
quadratic equation
You can eliminate choices A and D.
This is a quadratic equation.
Eliminate choice C.
Answer: The answer is B.
Check
This equation is in the form
Multiple-Choice Test Item
Which rule describes a linear function?
A
Answer: C
B
C
D
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Example 1 Graph Quadratic Functions
Example 2 Use a Function to Solve a Problem
Example 3 Graph Cubic Functions
Graph
.
Make a table of values, plot the ordered pairs, and connect
Answer:
the points with a curve.
x
(x, y)
–1.5
(–1.5, –4.5)
–1
(–1, –2)
–0.5
(–0.5, –0.5)
0.5
(0.5, –0.5)
0
(0, 0)
1
(1, –2)
1.5
(1.5, 4.5)
Graph
.
Answer:
x
(x, y)
–2
(–2, 3)
–1
(–1, 1.5)
0
(0, 1)
1
(1, 1.5)
2
(2, 3)
Graph
.
x
(x, y)
–2
(–2, 7)
–1
(–1, –4)
0
(0, –3)
1
(1, –4)
2
(2, –7)
Answer:
Graph each function.
a.
Answer:
Graph each function.
b.
Answer:
Graph each function.
c.
Answer:
Geometry The height of a triangle is 4 times its base.
Write a formula for the area and graph it. Find the
area of the triangle whose base is 3 units.
Words
The area of a triangle is equal to one-half the
product of its base and height.
Variables
.
Equations Area is equal to one-half
A
=
the product of its
base and height
The equation is
. Since the variable b has an
exponent of 2, this function is nonlinear. Now graph
Since the base cannot be negative, use only positive
values of b.
b
0
0.5
1
1.5
2
2.5
(b, A)
(0, 0)
(0.5, 0.5)
(1, 2)
(1.5, 4.5)
(2, 8)
(2.5, 12.5)
.
By looking at the graph,
we find that for a base of
3 units, the area of the
triangle is 18 square units.
Geometry The length of a rectangle is 3 times its
width. Write a formula for the area and graph it. Find
the area of the rectangle whose width is 3.5 inches.
Answer:
Graph
Answer:
.
x
(x, y)
–2
(–2, 4)
–1
(–1, )
0
(0, 0)
1
(1, – )
2
(2, –4)
Graph
.
x
(x, y)
–1.5
(–1.5, –4.75)
–1
(–1, 0)
0
(0, 2)
1
(1, 4)
1.5
(1.5, 8.75)
Answer:
Graph each function.
a.
Answer:
Graph each function.
b.
Answer:
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