Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to:

Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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Lesson 13-1 Lesson 13-2 Lesson 13-3 Lesson 13-4 Lesson 13-5 Lesson 13-6 Polynomials Adding Polynomials Subtracting Polynomials Multiplying a Polynomial by a Monomial Linear and Nonlinear Functions 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 10w 4 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 4 degree 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

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. Answer: The perimeter is Formula for the perimeter of a rectangle Replace with and w with Distributive Property Group like terms. Simplify.

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

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 each difference. a. Answer: b. Answer:

Find. Answer: The difference is x 17. To subtract (3x + 9), add ( 3x 9). Group the like terms. Simplify.

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. Answer: The length is than the width. Simplify. 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:

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 Plan You know the perimeter of the dog run. You want to find the dimensions of the dog run. 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

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 y 2 25 4 17 6 9 8 1 8 8 8 As x increases by 2, y decreases by 8. So, this is a linear function. Answer: linear

Determine whether the table represents a linear or nonlinear function. +3 +3 +3 x y 5 2 8 4 11 8 14 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 x 10 9 8 7 y 4 7 10 13 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 quadratic equation You can eliminate choices A and D. Answer: The answer is B. This is a quadratic equation. Eliminate choice C.

Check This equation is in the form

Multiple-Choice Test Item Which rule describes a linear function? A B C D Answer: C

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 the points with a curve. Answer: x 1.5 1 0.5 0.5 0 1 1.5 (x, y) ( 1.5, 4.5) ( 1, 2) ( 0.5, 0.5) (0.5, 0.5) (0, 0) (1, 2) (1.5, 4.5)

Graph. x 2 (x, y) ( 2, 3) Answer: 1 ( 1, 1.5) 0 (0, 1) 1 (1, 1.5) 2 (2, 3)

Graph. x 2 1 (x, y) ( 2, 7) ( 1, 4) Answer: 0 (0, 3) 1 (1, 4) 2 (2, 7)

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 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. b (b, A) 0 0.5 1 1.5 2 2.5 (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 2 1 (x, y) ( 2, 4) ( 1, ) 0 (0, 0) 1 (1, ) 2 (2, 4)

Graph. x 1.5 1 0 1 1.5 (x, y) ( 1.5, 4.75) ( 1, 0) (0, 2) (1, 4) (1.5, 8.75) Answer:

Graph each function. a. Answer:

Graph each function. b. Answer:

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