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Pre-Algebra Interactive Chalkboard
Copyright © by The McGraw-Hill Companies, Inc.
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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 becausehas a variable in the denominator.
Answer: This is a polynomial because it is the difference of two monomials. There are two terms, so it is a binomial.
Determine whether is a polynomial. If it is, classify it as a monomial, binomial, or trinomial.
Determine whether each expression is a polynomial. If it is, classify it as a monomial, binomial, or trinomial.
a.
b.
Answer: yes; trinomial
Answer: not a polynomial
Answer: The variable w has degree 4, so the degree
of –10w4 is 4.
Find the degree of .
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.
b.
Answer: 3
Answer: 8
Find the degree of .
04
7
degreeterm
Answer: The greatest degree is 7. So, the degree of the
polynomial is 7.
Find the degree of .
Answer: The greatest degree is 7. So, the degree of the
polynomial is 7.
7
4
degreeterm
Find the degree of each polynomial.
a.
b.
Answer: 5
Answer: 6
Answer:
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: 2
Area The formula for the surface area S of a cylinder with height h and radius r is .Find the degree of the polynomial.
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Example 1 Add Polynomials
Example 2 Use Polynomials to Solve a Problem
Find .
Method 1 Add vertically.
Method 2 Add horizontally.
Align like terms.
Add.
Associative and Commutative Properties
Answer: The sum is 10w + 1.
Find .
Method 1 Add vertically.
Method 2 Add horizontally.
Align like terms.
Add.
Write the expression.
Group like terms.
Simplify.
Answer: The sum is
Find .
Write the expression.
Simplify.
Answer: The sum is
Find .
Answer: The sum is .
Leave a space because there is no other term like xy.
Find each sum.
a.
b.
c.
d.
Answer:
Answer:
Answer:
Answer:
Geometry The length of a rectangle is
units and the width is 8x – 1 units.
Find the perimeter.
Answer: The perimeter is
Formula for the perimeter of a rectangle
Distributive Property
Simplify.
Group like terms.
Replace with
and w with
Answer: The length of the rectangle is 16 units.
Find the length of the rectangle if
Write the expression.
Replace x with –3.
Simplify.
Geometry The length of a rectangle is
units and the width is 6w – 3 units.
a. Find the perimeter.
b. Find the length if
Answer:
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 .
Answer: The difference is .
Align like terms.
Subtract.
Find .
Subtract.
Answer: The difference is .
Align like terms.
Find each difference.
a.
b.
Answer:
Answer:
Find .
To subtract (3x + 9), add (–3x – 9).
Group the like terms.
Simplify.
Answer: The difference is x–17.
Find .
Align the like terms and add the additive inverse.
Answer:
The additive inverse of
Find each difference.
a.
b.
Answer: 10c – 7.
Answer:
Geometry The length of a rectangle isunits. The width is units. How much longer is the length than the width?
Answer: The length is units longer than the width.
difference in measurement
Add additive inverse.
Group like terms.
Simplify.
Substitution
Profit The ABC Company’s costs are given bywhere 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 .
Answer: – 24x – 16
Simplify.
Distributive Property
Find .
Simplify.
Answer:
Distributive Property
Find each product.
a. 3(–5m – 2)
b. (4p – 8)(–3p)
Answer: –15m – 6
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
Answer: The width of the dog run is 6 feet, and the length is
Write the equation.
Replace P with 56 and
Combine like terms.
Distributive Property
Subtract 8 from each side.
Divide each side by 8.
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.
Answer: nonlinear Answer: linear
Determine whether each graph represents a linear or nonlinear function.a. b.
Determine whether represents a linear or nonlinear function.
Answer: This equation represents a linear functionbecause it is written in the form
Answer: This equation is nonlinear because x is raised to the second power and the equation cannot be written in the form
Determine whether represents a linear ornonlinear function.
Determine whether each equation represents a linearor nonlinear function.
a.
b.
Answer: nonlinear
Answer: linear
Determine whether the table represents a linear or nonlinear function.
x y
2 25
4 17
6 9
8 1
+2
+2
–8
–8
–8
As x increases by 2, y decreases by 8. So, this is a linear function.
Answer: linear
+2
Determine whether the table represents a linear or nonlinear function.
x y
5 2
8 4
11 8
14 16
+3
+3
+3
+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.
169
137
115
103
yxa. b.
137
108
79
410
yx
Answer: nonlinear 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
You can eliminate choices A and D.
This is a quadratic equation. Eliminate choice C.
Answer: The answer is B.
quadratic equation
CheckThis equation is in the form
Answer: C
Multiple-Choice Test ItemWhich rule describes a linear function?
A 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
Answer:
Graph .
(1.5, 4.5)1.5
(1, –2)1
(0, 0)0
(0.5, –0.5)0.5
(–0.5, –0.5)–0.5
(–1, –2)–1
(–1.5, –4.5)–1.5
(x, y)x
Make a table of values, plot the ordered pairs, and connect the points with a curve.
Answer:
Graph .
(2, 3)2
(1, 1.5)1
(0, 1)0
(–1, 1.5)–1
(–2, 3)–2
(x, y)x
Answer:
Graph .
(2, –7)2
(1, –4)1
(0, –3)0
(–1, –4)–1
(–2, 7)–2
(x, y)x
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 theproduct of its base and height.
Variables .
Equations Area is equal to one-half the product of its base and height
A =
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.
(2.5, 12.5)2.5
(2, 8)2
(1.5, 4.5)1.5
(1, 2)1
(0.5, 0.5)0.5
(0, 0)0
(b, A)b
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:
Answer:
Graph .
(2, –4)2
(1, – )1
(0, 0)0
(–1, )–1
(–2, 4)–2
(x, y)x
Answer:
Graph .
(1.5, 8.75)1.5
(1, 4)1
(0, 2)0
(–1, 0)–1
(–1.5, –4.75)–1.5
(x, y)x
Answer:
Graph each function.
a.
Graph each function.
b.
Answer:
Explore online information about the information introduced in this chapter.
Click on the Connect button to launch your browser and go to the Pre-Algebra Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.pre-alg.com/extra_examples.
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