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Pre-Algebra Interactive Chalkboard

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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:

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