Unit 6:

Polynomials

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Day 1 : Basics of Polynomials

Before we continue our study of functions, we will revisit a particular type of expression you

have worked with in previous math courses. These expressions are known as polynomials.

Consider the following table containing examples of expressions that are polynomials and

examples of expressions that are not polynomials.

Polynomial Not a Polynomial

4x2 + 8x – 1

5x – 3

y3 + 3y2 + y

1 – 2

4x

3x2y – 2xy + 14

9x2 + 2x + x

1

2x + 3x

74 2 x

1

12 xx

Exercise 1: Work with your group to identify at least two key characteristics of polynomial

expressions. Record your thoughts here:

Exercise 2: Consider the following set of expressions. Circle those expressions you believe are

polynomials.

5x3 + 2 12 xx 3 – x

8xy-2 + 4x 2x5 – x3+ 8x2 – 4x 234 xx

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Key Concept: A polynomial expression is any expression of the form

...21 nnn cxbxax constant , where the exponents, n, n 1, n 2 , etcetera are all

________________ integers. Note that not all powers need to be present because the

coefficients, i.e. a, b, c, etc. can be ________________.

Exercise 3: Examine the two polynomials shown here and answer the questions that follow.

A) x – 7x2 – 4 + 2x3 B) 2x3 – 7x2 + x – 4

a) How are polynomials A and B similar?

b) How are they different?

c) Are they equivalent expressions? Identify at least one way you could check to

see if they are equivalent.

Key Concept: It is often useful to place polynomials in their standard form. The

standard form of a polynomial is simply achieved by writing it as an equivalent expression where

the powers on the variables are in ___________________________ order.

Exercise 4: Write each of the following polynomials in standard form.

a) 3x2 + 5x3 + 7 – 8x b) 3 – 2x – 5x2

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Key Vocabulary:

1) A term is any combination of numbers and variables that are multiplied together.

ex)

2) A ________________ is an expression with a single term.

ex)

3) A ________________ is an expression with two terms added or subtracted together.

ex)

4) A _____________________ is an expression with three terms added or subtracted together.

ex)

5) A _____________________ is an expression with two or more terms that are added or subtracted

together.

ex)

6) The degree of a polynomial function is the largest exponent that the function contains.

7) The __________________ is a value in a polynomial that cannot be changed. Ex)

8) The __________________ is the number that is in front of a variable. Ex)

9) The _________________________________ is the number that is in front of the variable of

largest exponent.

Exercise 5:

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Polynomials are simply abstract representations of numbers that we see every day and they

behave like these numbers as well so we can complete various operations with them. Let’s look

at adding polynomials.

The key to addition whether it is adding numbers or polynomial expressions is to add, or

combine, _____________ ____________________.

Exercise 6: a) Find the sum of 5x2 – 3x + 2 and 3x2 – 4x – 1.

b) Find the sum of 7x3 – 3x2 + 5 and -2x3 + 6x2 – 5.

c) Write a binomial expression in standard form to represent the perimeter of the

following rectangle:

x – 4

x + 6

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Subtracting polynomials uses the same concept as subtracting numbers. You can subtract term

by term or make the problem easier by changing it into an equivalent addition problem since

subtracting is the same as adding the ___________________. Be careful and watch your signs!!

Exercise 7: a) (4x2 – 2x + 8) – ( 2x2 + 5x – 4)

b) From 5x2 + 3xy + 6y2 subtract 4x2 + 3xy + 2y2.

c) Subtract from . Express the result as a trinomial.

d) When 5x + 4y is subtracted from 5x – 4y, the difference is

1) 0

2) 10x

3) 8y

4) -8y

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Exercise 8: Mixed Practice with Addition and Subtraction

a) Create an expression so that, when it is added to 3x – 7, it equals 0.

b) A triangle has a perimeter of 2x2 – 10 x + 8. The lengths of two of the sides

are x2 + 3x – 1 and x + 5. What is the length of the third side?

c) A company produces x units of a product per month, where C(x) represents

the total cost and R(x) represents the total revenue for the month. The

functions are modeled by C(x) = 300x + 250 and R(x) = -0.5x2 + 800x – 100.

The profit is the difference between revenue and cost where P(x) = R(x) –C(x).

What is the total profit, P(x), for the month?

1) P(x) = -0.5x2 + 500x – 150

2) P(x) = -0.5x2 + 500x – 350

3) P(x) = -0.5x2 – 500x + 350

4) P(x) = -0.5x2 + 500x + 350

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Day 2 : Multiplying Polynomials

Just as polynomials can be added and subtracted to create new polynomials, polynomials can also be

multiplied to create new polynomials. In order to multiply polynomials, we need to be aware of and

able to use some helpful rules about exponents.

Let’s review two basic but very important exponent rules.

Recall that an exponent is a way to indicate repeated ___________________ by the same number or

variable.

Exercise 1: Write out what each of the following terms involving exponents means as an extended

product.

a) 5x3 b) x2y4 c) (3x)2 d) (9x2)3

Exercise 2: Write out each of the following products and then express them in the form xn .

(a) x2x3 (b) x5x2 (c) x4x4

So, what’s the pattern? Can you give a generic rule for what happens when we multiply

two terms that have the same base?

Exercise 3: Quickly write each of the following products as a variable raised to a single power.

(a) x4x9 (b) x2x3x4 (c) 4y2y6

ba xx

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The rule we just found is helpful when multiplying monomials. Consider the following problem:

(2x3)(3x4)

Because of the _______________________ property, it is correct to say 2x3∙3x4 = 2∙3∙x3∙x4

Using what we know about multiplying integers and multiplying terms with the same bases we

get:

2x3∙3x4 = 2∙3∙x3∙x4 = _________________

We can also find a rule for simplifying the expression (xa)b .

Exercise 4: Try the following questions and see if you can find the pattern that helps simplify

this type of expression.

a) Rewrite the following terms as extended products and then express them in the form xn.

(x3)4 (x2)3

b) Make a general rule for all terms in the form of (xa)b:

bax )(

Exercise 5: What if there is a coefficient other than one? Consider the expression (2x5)3.

a) Rewrite the term as an extended product and then simplify it.

b) Can you identify a “short cut” or rule? Explain.

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Key Concepts: 1) When multiplying terms, multiply the coefficients but __________ the exponents.

ex) (2x2)(3x3)

2) When a term is being raised to an exponent we raise the coefficient to the

power but we _____________________ the exponents.

ex) (2x4)3 23x3*4

Exercise 6: Make sure you pay attention to details and use the appropriate rules when

simplifying expressions. Simplify the following expressions:

7x2 + 9x2 7x2 ∙ 9x2

Practice: Simplify the following expressions.

1) (-3y2)3 2) (-2x2)(5x4) 3) (-3a2b3)(-4a4b)

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4) (2x5y3)(4x3y2) 5) 4x2 – 7x2 6) (m2n3)4

7)

(3x2y)2 8)

(3m3)2

9)

(2n2)3 10) 43)( cy

11) (2a2b)(4ab2) 12) (6x2)(-3x5)

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Multiplying Polynomials

When two or more polynomials are multiplied, the resulting expression is also a polynomial. A

useful tool that can be used in the visualization and development of polynomial multiplication is

an area model.

Exercise 1: a) Determine the area of the following rectangles.

x 10

x x

Area: _________________ Area: _________________

x 10

x Area: ____________________

b) Write the following product as a binomial in standard form:

x(x + 10) =

c) What property did you use?

d) Multiply the following expressions using an area model. Be sure to express your final answer

as a polynomial in standard form.

i) 3(x + 3) ii) x(x – 4) iii) 4x(2x + 8)

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x 3 x -4

x

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iv) 2g2(3g2 – g + 8) v) (x + 2)(x – 3)

Exercise 2: a) Set up and use an area model to multiply (2x – 3)(x + 5). Express your answer as

a trinomial in standard form.

b) Without using an area model, use the distributive property to rewrite the

following product as a trinomial.

(2x – 3)(x + 5)

c) Which method do you prefer? Why?

2g2

x 2

x

-3

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Exercise 3: a) Set up and use an area model to multiply (x + 4)(x2 + 2x – 1)? Express your

answer as a polynomial in standard form.

b) Without using an area model, use the distributive property to rewrite the following

product as a polynomial in standard form.

(x + 4)(x2 + 2x – 1)

Exercise 4: Consider any of the multiplication problems completed so far in this section. What

strategies do you have to test for equivalence between the original product and your resulting

polynomial? (i.e. how can you check to see if your answer is correct?)

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Day 3 : Polynomial Arithmetic

1) What is the coefficient of the x term in the trinomial that is equivalent to (x – 3)(4x + 7)?

2) What is the value of the constant term when the product (2x – 8)(3x + 7) is written as a

trinomial?

3) If the difference is multiplied by , what is the result, written

in standard form?

4) Express the product of and in standard form.

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5) In the accompanying diagram, the width of the inner rectangle is represented by x – 2 and its

length by x + 4. The width of the outer rectangle is represented by 2x + 3 and its length by

2x - 3.

2x + 3

x - 2 2x - 3

x + 4

a. Write an expression to represent the area of the larger rectangle. Express the answer as a

binomial.

b. Write an expression to represent the area of the smaller rectangle. Express the answer as

a binomial.

c. Express the area of the unshaded region as a polynomial is terms of x.

6) Fred is given a rectangular pieces of paper. If the length of Fred’s piece of paper is

represented by 2x – 6 and the width is represented by 3x – 5, then the paper has a

total area represented by

(1) 5x – 11 (3) 10x – 22

(2) 6x2 – 28x + 30 (4) 6x2 – 6x – 11

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7) Which of the following expressions is equivalent to (x + 7)2?

(1) x2 + 49 (3) x2 + 14

(2) x2 + 7x + 49 (4) x2 + 14x + 49

8) When (2x - 3)2 is subtracted from 5x2, the result is

(1) x2 - 12x - 9 (3) x2 + 12x - 9

(2) x2 - 12x + 9 ( 4) x2 + 12x + 9

9) Which trinomial is equivalent to ?

1) 2) 3) 4)

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Practice: Simplify the following expressions.

1. (2x3y5) (-3x4y7) 2. (-3x3)4

3. (3x + 1)2 4. (x + 5)(3x2 – 4x – 7)

5. (x – 7)(x – 2) 6. (x + 4)(x – 6)

7. (3x + 1)2 8. (2x + 1)(2x2 – 3x + 1)

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9) Simplify the expression 2(x + 4)2 – 3(x – 1) . Express your answer as a trinomial in standard

form.

10) If the sum of 3x2 + 5x – 4 and –x2 + 3x – 2 is multiplied by , what is the result, written

in standard form?

11) In the accompanying diagram, the width of the inner rectangle is represented by x – 3 and

its length by x + 3. The width of the outer rectangle is represented by 3x + 4 and its length by

3x – 4.

3x + 4

x - 3 3x – 4

x + 3

a. Write an expression to represent the area of the larger rectangle. Express the answer as a

binomial.

b. Write an expression to represent the area of the smaller rectangle. Express the answer as a

binomial.

c. Express the area of the unshaded region as a polynomial is terms of x.

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Day 4 : Factoring Using GCF’s

Exercise 1: Consider the following area models and fill in the blanks with the missing terms:

Exercise 2: Answer the following questions using your prior knowledge:

a) What is a factor?

b) What are the factors of 12?

c) What are the factors of 36?

d) What is a prime number?

e) What is a composite number?

x2 4x

x

2x

2x2 5x

- 4x

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f) What is a greatest common factor for two or more given numbers?

g) What is the greatest common factor of 18 and 24?

h) What is the GCF of 64 and 144?

i) What is the GCF of 4x and 8x2?

Practice: Find the greatest common factor of the following.

1) 12v2, 30v5 2) x3y2, xy4

3) 7m, 21m2 4) 80x3, 30yx2

5) 36xy3, 24xy2, 8x2y2 6) 105x, 30x2, 75

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Factoring an expression means to rewrite the expression as an equivalent product of two or

more factors.

Exercise 3: Consider the expression 3x2 + 12x.

a) Determine the greatest common factor of the two terms.

b) Complete the area model using the GCF from part a as the width of the rectangle.

c) Use your work from part b to fill in the blanks:

3x2 + 12x = ______ ( _______ + _______ )

d) What property is evident in part c?

Exercise 4: Write the following expressions as the product of the polynomial’s greatest

common factor with another polynomial. Use of an area model is optional.

a) 4x2 + 8x – 24 b) 6g4 – 2g2

c) 36xy3 + 24xy2 – 8x2y2 d) x2 – 6x

3x2 12x

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Exercise 5: If 3x is one factor of 3x2 – 9x, what is the other factor?

(1) 3x (3) x – 3

(2) x2 – 6x (4) x + 3

Exercise 6: A rectangle has an area that can be modeled by the expression 4x2 – 8x + 2. Given

the width of the rectangle is 2, find an expression that represents the length of the

rectangle.

Exercise 7: Sometimes greatest common factors are more complicated than simple monomials.

Rewrite each of the following expressions as the product of two binomials by

factoring out a common binomial factor.

a) 2x(x + 5) + 4(x + 5) b) (y + 4)(y + 7) – (y – 2)(y + 4)

c) (x + 7)(x – 1) + (x + 7)(2x – 3) d) (2x – 1)(2x + 8) – (2x – 1)(x – 3)

Exercise 8: Which of the following is equivalent to the expression (x – 3)(2x + 7) – (x – 3)(x – 4)

(1) (x – 3)(x + 3) (2) (x – 6)(x + 10)

(3) (x – 3)(x + 11) (4) (x – 4)(x – 6)

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Key Concept: The word factor has two meanings and both are important.

1. Factor (verb): To rewrite an algebraic expression as an equivalent ___________________.

2. Factor (noun): An algebraic expression that is one part of a larger factored expression.

1) Sometimes an expression does not have a greatest common factor. That does not mean the

expression cannot be factored. For instance, the expression x2 + 5x + 6 can be factored and

written as (x + 2)(x + 3).

a) Verify that the following statement is true:

x2 + 5x + 6 = (x + 2)(x + 3)

b) Discuss how we might find the factor a trinomial with no GCF.

2) Factor the following trinomial x2 +6x + 8.

3) Factor the following trinomial x2 – x – 12.

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Day 5: More Factoring Fun

Factor the following binomials:

1. x2 + 9x 2. 4x2 – 12x

3. 7x – 56 4. 12x – 36

5. Factor the trinomial x2 – 8x + 15. If necessary, use the area model below.

a) Describe how to factor x2 – 8x + 15 without the area model.

6. Rewrite the following in factored form using the method you prefer.

a.) x2 – 7x + 10 b.) x2 + 8x + 12 c.) x2 – 5x – 24

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7. Now try to factor x2 – 36

a. Expressions in the form a2 – b2 are sometimes referred to as the difference of

perfect squares. Why?

8. Rewrite the following in factored form.

a.) 9x2 – 16 b.) 81 – 4y2 c.) x4 – 36

Hierarchy of Factoring

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Exercise 1: Factor the following expressions:

***Remember when factoring you must look for a GCF first before performing any other method***

x2 – 81 x2 + 9x x2 + 9x + 8

4x4 – 121 24x2 – 12 x2 – 5x + 6

x6 – 25 x2y3 – 8xy2 x2 – 2x – 8

49 – x2 4x2 – 12x x4 + 3x2 – 18

Exercise 2: Complete the following Regents practice questions.

a) Which expression is equivalent to ?

1)

2)

3)

4)

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b) If the area of a rectangle is expressed as , then the product of the length and

the width of the rectangle could be expressed as

1)

2)

3)

4)

c) Which expression is equivalent to x4 – 12x2 + 36?

(1) (x2 – 6)(x2 – 6) (3) (6 – x2)(6 + x2)

(2) (x2 + 6)(x2 + 6) (4) (x2 + 6)(x2 – 6)

Exercise 3: Kevin, Daryl, and Katie each factored the expression 49 – 16y2.

Kevin says the correct answer is (7 + 4y)(7 – 4y).

Daryl says the correct factorization is (4y + 7)(4y – 7).

Katie says the correct answer is -1(4y + 7)(4y – 7).

Who is correct? Explain.

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Exercise 4: A rectangle has dimensions as shown below in terms of an unknown variable, x.

Find a binomial expression for the length of the rectangle in terms of x. Justify your answer

based on the expressions for the rectangle’s length and area.

Exercise 5: Describe at least two different ways you can check your answer to make sure your factored

expression is equivalent to the original expression.

Area = x2 + 34x – 72 Width = x – 2

Length = ?

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Day 6: Factoring Completely

Exercise 1: Ted, Fred, and Jed were given the expression 3(x – 2)(x + 5) in factored form and were

asked to simplify the expression by multiplying. Below is the work each student completed.

Ted: Fred: Jed:

3(x – 2)(x + 5) 3(x – 2)(x + 5) 3(x – 2)(x + 5)

(3x – 6)(x + 5) (3x – 6)(3x + 15) 3(x2 + 5x – 2x – 10)

3x2 + 15x – 6x – 30 9x2 + 45x – 18x – 90 3(x2 + 3x – 10)

3x2 + 9x – 30 9x2 + 27x – 90 3x2 + 9x - 30

Whose answer is correct? Why?

Exercise 2: Kate and Jenny were asked to factor the expression 2x2 – 50 and got two different answers.

Here are their answers:

Kate: 2(x2 – 25) Jenny: 2(x + 5)(x – 5)

a) Check to see if these students are correct by multiplying out their factored expressions.

2(x2 – 25) 2(x + 5)(x – 5)

b) Whose factors are correct?

c) If Kate and Jenny had been told to “factor completely”, whose answer do you think would be

better? Why?

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***Remember when factoring you must look for a GCF first before performing any other method***

Practice: Factor the following completely.

1) 2x2 + 14x + 20 2) 4y2 + 24y + 20

Exercise 1: Some expressions cannot be written as the product of two or more factors. Here are two

examples: x2 + 4 and x2 + x + 1.

a) How can you verify that these expressions are not factorable?

b) The only factors these expressions have are 1 and itself. What do we call numbers whose

only factors are 1 and itself?

Key Concepts:

An expression is factored completely when _________________________________

_____________________________________________________________________.

An expression that has no factors other than 1 and itself is ______________________.

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Factor the following trinomials:

1.) 2x2 – 5x – 3 2.) 4x2 – x – 3

An Alternative Method for Factoring: Factoring By Grouping (“Splitting the Middle”)

4x2 – x – 3

1st – Look for any common factors first!

2nd – Find ca

3rd – Find two new factors of ca that add up to b

4th - Split the middle term into two terms using the sum

of the two new factors, including the proper signs.

5th – Group the four terms into two pairs

6th – Factor each pair by finding the GCF

7th – Factor out the common binomial

3) 2x2 + 9x + 4 4) 2x2 – 7x – 4

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5) 3a3 + 18a2 + 15a 6) 2x2 – 28x + 98

7) 3n2 – 2n – 5 8) 5x2 + 19x – 4

9) 8x2 – 72 10) 5x4 – 5x2

11) c3 + 17c2 + 16c 12) 50 – 2x2

13) 12x2 – 72x – 84 14) a3 – a

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15) y4 – 81 16) x5 – 4x4 – 21x3

17) When factored completely, the expression p4 – 81 is equivalent to

(1) (p2 + 9)(p2 – 9)

(2) (p2 – 9)(p2 – 9)

(3) (p2 + 9)(p + 3)(p – 3)

(4) (p + 3)(p – 3) (p + 3)(p – 3)

18) Four expressions are shown below.

I

II

III

IV

The expression is equivalent to

1) I and II, only

2) II and IV, only

3) I, II, and IV

4) II, III, and IV

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Day 7: Factoring Practice/Review

Exercise 1: Factor the following completely: (only 1 of the 6 problems is “prime”)

a) x2 + 5x + 4 b) x2 + 14x + 48

c) x2 – 25 d) x2 + 25

e.) x2 – 4x f.) -2x2 + 5x – 2

Practice:

1.) 2x2 + 7x – 7 2.) 3k2 − 10k + 7

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3) 2t2 + 13t + 15 4.) 9x2 – 3x – 12

5) One factor of the expression 7p2 – 20p + 12 is (p – 2). What is the other factor?

_____ 6) 4x2 + 12x + 9 is equivalent to

(A) (2x – 3)2 (C) (2x + 3)(2x – 3)

(B) (2x – 9)(2x + 1) (D) (2x + 3)2

______ 7) Factored completely, the expression 12x4 + 10x3 – 12x2 is equivalent to

(A) x2 (4x + 6)(3x – 2) (C) 2x2(2x – 3)(3x + 2)

(B) 2(2x2 + 3x)(3x2- 2x ) (D) 2x2(2x + 3)(3x – 2)

8) Emily says the expression 8x2 + 20x + 8 is factored completely when it is in the form 4(2x2 + 5x + 2).

Do you agree or disagree? Explain.

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9) The area of a rectangle is given by the expression 7m2 + 6m – 1. Determine the dimensions of the

rectangle in terms of m.

A = 7m2 + 6m – 1

10) Two mathematicians are neighbors. Each owns a separate rectangular plot of land that

shares a boundary and has the same dimensions. They agree that each has an area of

𝟐𝒙2 + 𝟑𝒙 + 𝟏 square units. One mathematician sells his plot to the other. The other

wants to put a fence around the perimeter of his new combined plot of land. How many

linear units of fencing will he need? Write your answer as an expression in 𝒙.

( This question has two correct approaches and two different correct solutions. Can you find them both?)

11) Factor: 2 6 9

4 16r r

(Around Room Activity)