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MATHEMATICS 10C

UNITS: P1-2

Operations with Polynomials

Prepared by theMathematics Department

Bishop Carroll High SchoolFeb, 2011

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MATHEMATICS 10C

Operations with Polynomial Expressions

INFORMATION

Title: Operations with Polynomial Expressions

Prerequisite: Grade 9 level exponents (Whole number exponents)

Materials Required: Foundations and Pre-Calculus Mathematics 10, Math XL for

SchoolEvaluation: Test will follow Units P1 2.

INTRODUCTION

The knowledge of polynomials will allow you to use algebra to solve many types of real

world problems. How do I maximize profit in my small business? What would be the best way

to fence my ranch? What quantity of stocks do I need to buy at the present lower rate to lower

my overall break-even point? Of course, to solve these types of problems we must start at a

much more basic level.

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OUTCOMES

Upon completion of this unit a student will be able to:

1. Define polynomials and demonstrate an understanding of the components of the

definition they relate to the simplification, expansion and factoring of polynomials.

2. Demonstrate an understanding of the algebraic method of simplifying polynomial

expressions (limited to monomials, binomials and trinomials) using the operations of

addition, subtraction and multiplication.

If you have any questions or difficulties, please ask for assistance in the Math Resource Centre.

LEARNING ACTIVITIES Carefully read through this learning guide and complete all exercises and

activities.

Please be sure to get help whenever you need it.

Each learning activity may contain text book references and alternative

activities to help you master the outcomes.

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The following icons will flag different types of learning activities:

Icon Explanation of Icon

Refer to your textbookThe two icons to the right will further indicate animations,or dynamic activities available in the web based ebook.

HAND OUT: This activity requires a hand out. It is available onD2L or in the Math Resource Centre

Go to D2L to find the content or activity.

NOTEBOOK OR WORKBOOOK: Do the exercises in yourworkbook or notebook.

LEARNING STRATEGIES TIPS: Throughout the learning guide,useful learning strategies and study skills tips are provided thatcan be applied to all subject areas to help you become an efficient,effective learner.

Log in to Math XL and do the Homework and/or quiz. Rememberthese are for practice only and are not included in your mark.

LINK: Follow the internet links. New links may have beenadded since the printing of this learning guide. Always checkon D2L for the latest learning guide to see if there are newlinks.In a Nutshell Online Seminar: Videos of each concept in the unitare on D2L under content.

Mathematics Glossary:An excellent resource for help or clarification at

any time during this learning guide. Many of the definitions are animatedand interactive. If you do not see the word under your grade level,try Grades 7-9. You can also find this glossaryon the Learn Albertawebsite using the hyperlink,Mathematics Glossary.
http://www.learnalberta.ca/content/memg/index.htmlhttp://www.learnalberta.ca/content/memg/index.htmlhttp://www.learnalberta.ca/content/memg/index.htmlhttp://www.dreamstime.com/royalty-free-stock-photography-lightbulb-illustration-image12670647http://www.dreamstime.com/royalty-free-stock-photo-handwriting-formulas-image1858755http://www.dreamstime.com/royalty-free-stock-images-a-stack-of-textbooks-on-white-image11917389http://www.learnalberta.ca/content/memg/index.html

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I. PREREQUISITE SKILLS

Review of the Language of Algebra and Polynomial Definition

In previousmath courses you studied some of the concepts and operations with

polynomials. Before we begin to learn more in this unit lets review.

Algebra is a branch of mathematics that uses symbols to represent unknown numbers

or quantities.

Examples: 1. The value of an unknown number of dimes, x, is 10x.

2. The number of boys, y, and girls, x, in a class of 30 students isy + x = 30.

3. The area, A, of a rectangle of length, l, and width, w, is A = lw.

A term is an expression formed from the product of numbers and/or variables.

Example 10x 3 has 2 terms: 10x and -3

10 is a numericalcoefficient, x is the variable and -3 is the constant term

II. DEFINE A POLYNOMIAL

Memorizing the definition of a polynomial is not sufficient. We must understand each of

the components of the definition. These components were discussed in part I. When you put

the concepts of part I together you have the knowledge to form a definition for polynomials and

a greater understanding of what polynomials are and how they work.

In algebra unknown parts of a number are represented using variables.

When we use variables, we are creating algebraic expressions

Algebraic expressions are made up of different parts called terms

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Lets build a polynomial for the following scenario.

Bobs cellphone company charges him $10 a month plus 45 cents per minute for each minutehe goes over 30 minutes.

Let C = the monthly cost of his cellphone. Let t = time in minutes

Using the variables, how would you calculate his monthly cost?

He pays $10 + 45 cents per minute for each minute over 30.

C = $10 + 0.45(minutes used per month 30 minutes)

C = $10 + 0.45(t-30)

C = 10+0.45(t-30)

The right side of this formula is the polynomial $10+0.45(t-30).

It represents the calculations Bob must use to calculate his cost.

When he knows how many minutes he has used he can calculate his cost. (for now

he uses the variable t to represent the unknown time)

When he does this calculation he will get a number.

That number is the cost of his cellphone for the month.

Therefore the polynomial $10+0.45(t-30) represents the number that is his cellphone bill.

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The definition for polynomials is not in your textbook. You may have already looked up

the definition in the Mathematics Glossary. This definition may appear a bit different than the

one you just developed. It is not. The mathematics glossary also provides an applet and

examples of algebraic expressions that are polynomials and examples of algebraic expressions

that are not polynomials.

All of these are useful, however, determining what is not a polynomial is not part P1-

You will study this concept in more detail in R1-3 and your grade 11 math course. You will also

see it mentioned in some of the online review lessons you are given a link to. It is not difficult

do not let this deter you from doing those lessons.

For this unit it is important that you remember that:

All algebraic expressions are not polynomials

All terms in polynomials are formed using the product of numbers and/or

variables

If you wish to explore the definition and why some algebraic expressions are not polynomials

click here.

Please note: The review section of this learning guide is not in your textbook. Therefore it is

important that you follow the links to see additional examples and get practice. Please see ateacher in the math area to get help with the review activities before moving on to the new

material.
http://c/Users/me/Desktop/poly/WHY%20SOME%20ALGEBRAIC%20EXPRESSIONS%20ARE%20NOT%20POLYNOMIALS.docxhttp://c/Users/me/Desktop/poly/WHY%20SOME%20ALGEBRAIC%20EXPRESSIONS%20ARE%20NOT%20POLYNOMIALS.docxhttp://c/Users/me/Desktop/poly/WHY%20SOME%20ALGEBRAIC%20EXPRESSIONS%20ARE%20NOT%20POLYNOMIALS.docx

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III. CLASSIFICATION OF POLYNOMIALS

Polynomials can be classified in two ways. According to the number of terms they

contain, or according to their degree. Read the following examples to see how to this is done,

then do the practice exercise.

NAMING POLYNOMIALS BY THE NUMBER OF TERMS THEY CONTAIN

1 term is a monomial example 3x

2 terms is a binomial example 5x 2

3 terms is a trinomial example 2x2 x + 1

4 or more terms is a polynomial example 7x3 5x2 x + 1

NAMING POLYNOMIALS BY DEGREE

The degree of a term is equal to the exponent of the variable if there is only 1 variable

and it is the sum of the exponents if there is more than 1 variable.

Example 4x5 is degree 5

7x2y is degree 3 (2 + 1 = 3)

The degree of a polynomial is equal to the degree of the term with the highest degree.

Example 8x2y 5xy + 3 has 3 terms

degree of each term

The degree of the polynomial is 3.

Hippo Campus Follow the link to do a review lesson and get practice on liketerms and the degree of a polynomial by completing lesson 1 and byselecting homework under the lesson title.If you have trouble using this website a video is provided in D2L undercontentInstructional videos for websites

3 2 0
http://www.montereyinstitute.org/courses/Algebra%20IB/course%20files/multimedia/lesson22/lessonp.htmlhttp://www.montereyinstitute.org/courses/Algebra%20IB/course%20files/multimedia/lesson22/lessonp.htmlhttp://www.montereyinstitute.org/courses/Algebra%20IB/course%20files/multimedia/lesson22/lessonp.html

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Try the Practice exercise below. Make sure you check your answers before moving on.

EXERCISE 1(a) Classify the following polynomials as monomials, binomials, etc.

(b) State the degree of the polynomial.

1. x + 6 6. 9abc 3ab + 4abc

2. 6a4 + 9a3 + 4 7. -12a4b

3. 4m 8. 3x5 4x4 + 9x3 6x + 4

4. -5 9. 19

5. 10a3bc 10. 4xy 3xy2 +9x2 y4

EXERCISE 2

1. Identify the variable in the following polynomial expression.

a. 142 2 xx

b. 14103 2 bb

c. 695 2 qq

2. Identify the coefficients in the following polynomial expressions.

a. 2058 2 xx

b. 7124 2 xx

c. 1362 2 nn

3. Identify the constant term in each of the following polynomial expressions.

a. 25105 2 xx

b. 2197 2 nn

c. 822 aa

Variable: a letter or symbol

representing a quantity in a

polynomial expression.

Eg.

The variable is x.

Constant Term: a term that

contains no variable.

Eg.

The constant is6. (note the

negative is part of the

constant term).

Coefficient: the numerical

factor of a term.

Eg.

The coefficients are 3 and -4

(note the negative is part of

the coefficient).

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Solutions

EXERCISE 1

1. (a) binomial (b) 2

2. (a) trinomial (b) 4

3. (a) monomial (b) 1





8. (a) polynomial in 5 terms

(b) 5



EXERCISE 2

1. a) the variable is x 2. a) 8, 5 3. a) 25

b) the variable is b b) -4, 12 b) -21

c) the variable is q c) -2, -6 c) 8

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IV SIMPLIFYING POLYNOMIAL EXPRESSIONS Review

A RULES FOR THE ORDER OF OPERATIONS

The Order of Operations are the commandments of mathematics: they are always in

force and never, ever vary! Polynomials represent numbers. Therefore the Order of

Operations applies to all of your work with polynomials and will make simplification easier!

Step1. Do calculation within brackets.

Step 2. Then calculate powers.

Step 3. Then do all division and multiplication in the order in which they appear from left to

right.

Step 4. Then do the addition and subtraction in the order in which they appear from left to right.

Sometimes the acronym BEDMAS is used to help remember these rules:

B - brackets

E - exponents

D - division

M - multiplication

A - additionS subtraction

Example Calculate: 4 x 6 (2 + 2)

= 4 x 6 (4 + 4)

= 4 x 6 (8)

= 16 x 6 (8)

= 16 x 6 8

= 96 8 = 12

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The negative sign

outside of ( )

changes the sign of

all terms in the ( )!

B LIKE TERMS

Algebraic terms that have the same variables and the same exponents connected to those

variables are called like terms. Otherwise they are unlike terms.

Example 5xy -2xy and xy are like terms

xy, the variables and their exponents, of each term is identical

3axy, 4xy, and 2xz are unlike terms. The variables in each term are different.

2x2y2, x2y, 3xy2 are unlike terms because even though the variables are the

same, their exponents differ.

C ADDITION OR SUBTRACTION OF POLYNOMIALS

To add terms in one or more polynomials requires that we have like terms. Liketerms are

terms with the same variables and the same exponents only numerical coefficients differ.

To add or subtract like terms, you simply add or subtract the numerical coefficients.

Example 1: 2xy + 4xy = (2 + 4)xy

= 6xy

Example 2: Simplify (3x + 2y) (2x 5y)

Solution: = (3x + 2y) (2x 5y)

Remember -(2x 5y) = -2x + 5y

= 3x + 2y 2x + 5y

= x + 7y

Since x + 7y are unlike terms you cannot simplify further.

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Sometimes you may find it easier to rearrange the expression so like terms are together.

Rearranging the order does not change the value of the polynomial.

Example 3: (3x 3) + (-x + 4)

= 3x x 3 + 4

= 2x + 1

Example 4: (3x 4) (2x + 3)

= 3x 4 + -2x + -3

= 3x 2x 4 3

= x 7

Alternate Method

The vertical method is way or organizing the addition or subtraction of polynomials.

Example 5: (x2 2x 3) (2x2 + 3x 1)

= (x2 2x 3)

(2x2 + 3x 1)

Hippo Campus Do lessons 1 and 2 to review the definition of like terms andsee more examples of multiplication and do an interactive lesson. To practice

these concepts, click on the word homework under the lesson title. If youhave trouble using this website a video is provided in D2L under contentInstructional videos for websites.

Look under content In a Nutshell videos in D2L to view the online seminarcalled Adding and Subtracting Polynomials.

Follow the link to play a game which will give you practice in simplifyingpolynomials. Make sure you read the instructions on the first webpage soyou can adjust the speed of the game. Have Fun!

Complete exercise 3 in your notebook. Check your work before continuing.If you need assistance please ask a teacher.

Careful! The subtraction sign infront of the parentheseschanges the sign of all termsinside the parentheses!
http://www.montereyinstitute.org/courses/Algebra%20IB/course%20files/multimedia/lesson22/lessonp.htmlhttp://www.slidermath.com/rpoly/Polya3.shtmlhttp://www.montereyinstitute.org/courses/Algebra%20IB/course%20files/multimedia/lesson22/lessonp.htmlhttp://www.montereyinstitute.org/courses/Algebra%20IB/course%20files/multimedia/lesson22/lessonp.html

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EXERCISE 3Simplify the following polynomial expressions by combining like terms.

1. (2a 3b) + (a 2b)

2. (6n 2m + 4p) (m 2n + 6p) + (5p + m n)

3. (a

2ab + b) + (a + 2ab + b)4. (3x + b) (6x + 5b + 6)

5. (12abc + 6ab + 8) (10abc + 12ab 20)

6. (4xyz + 6xy) (2xyz 2x + 2y) + (-2xyz + 4x + 2y 4xy)

7. {4x (3y 2x + 2y)} {3x + 7y}

8. (3ab + 4b) (3a 5ab)

Solutions to Exercise 3

1. 3a 5b 5. 2abc 6ab + 28

2. 7n 2m + 3p 6. 2xy + 6x

3. 2a + b + b 7. 3x 12y

4. -3x 4b 6 8. 8ab + 4b 3a

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D POWERS

A brief review of the rules for working with powers is necessary before we beginmultiplication.

x4 4 is called the exponent

x is called the base

x4 is called a power and can be written as

x4 = x x x x Here in this example the exponent 4 indicates the number of

times x is used as a factor.

If two powers having the same base are multiplied, the exponents are added.

Example 1: x4 x3 = x4+3 = x7 (Product law of exponents)

Example 2: (xy)(xy4

) = (x2+2

)(y1+4

) = x4

y5

If two powers having the same base are divided, the exponents are added.

Example 3: x7 x4 = x7-4 = x3 (quotient law of exponents)

Hippo Campus Go to the link for lessons and practice that will review exponent

laws. After completing each lesson you can select homework under each lesson

title to get somepractice with instant feedback. If you have trouble using this

website a video is provided in D2L under content Instructional videos forwebsites.

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E MULTIPLYING POLYNOMIAL EXPRESSIONS

When multiplying 2 or more monomials, we multiply the numerical coefficients together, the

variables together, and put these two answers together. (Remember there is a multiplication sign

implied between the coefficients and the variables)

Example 1: (3x)(4x3)

Solution: (3x)(4x3) = (3)(4)(x)( x3)

= 12x2+3

= 12x5

Example 2: (6abc)(2ab)

Solution: (6abc)(2ab) = (6)(2)(a)(a)(b)(b)(c)

= 12(a1+2

)(b1+2

)(c)= 12a3b3c

Expand means to multiply

If two factors with like signs are multiplied the product is positive.

If two factors with unlike signs are multiplied the product is negative.

When multiplying more than 2 terms watch for the number of negative signs.

If there is an odd number of negative signs, the product is negative.

If there is an even number of negative signs, the product is positive.

Examples: (3x)(4x) = 12x

(-2x)(-3x) = 6x

(-2x)(+4x) = -8x

(+2x)(-5x)(-4x) = 40x3

(-2x)(-5x)(-4x) = -40x3

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F DIVIDING POLYNOMIAL EXPRESSIONS

To divide a monomial by a monomial, you divide the numerical coefficients and then divide

the literal coefficients. Remember that you will have to use the quotient law of exponents.

Study the following examples:

Example 1: 36a3b 4ab

Solution: 36a3b 4ab =3 2

36a b

4abHere you divide 36 by 4 and

a3b by ab.

= 9a3 -1

b2 -1

Note: quotient law of exponents is used.= 9ab

Example 2: -12x3y5 4xy

Solution: -12x3y5 4xy =3 5

2

-12x y

4xy

= -3x3-1y5-2= -3xy3

Example 3: (3xy)(8xy) 6xy

Solution: (3xy)(8xy) 6xy =2 2 2

2

(3x y)(8x y )

6xy

=4 3

2

24x y

6xy

= 4x3y

To divide a polynomial by a monomial, each term of the polynomial is divided by the

monomial.

Division rewritten as a fraction is

sometimes easier to interpret

correctly.

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Example 1: (18x 4x) 2x

Solution: (18x 4x) 2x =2

18x 4x

2x

Divide each term of the polynomial by 2x

=2

18x 4x

2x 2x

= 9x 2

Look under content in D2L In a Nutshell Videos view the In a nutshellonline seminar called Multiplying and Dividing Polynomials.

Hippo Campus Do lessons 2 and 3 listed in the menu on the left at this link.To practice the concept, click on the word homework under the lesson title.If you have trouble using this website a video is provided in D2L undercontentInstructional videos for websites.

Complete exercise 4 in your notebook. Check your work before continuing.If you need assistance please ask a teacher.

EXERCISE 4

1. (-4x2y

3) (2xy) 6. (3x

2 6x + 12) 3

2. (-25x2y) (-5xy) 7. (2x3 + 6x + 8x) 2x

3. (20x3y2) (30x3y2) 8. (5x + 4x) 20x

4. (4x2y)(5xy

3) (20xy

4) 9. (4x

3y 8x) 4x

5. (6x 8x) 2x

SOLUTIONS EXERCISE 4

1. -2xy2

2. 5x

3.2

3

4. x2

5. 3x4

6. x22x + 4

7. x2

+ 3x + 4

8.x 1

4 5

9. x2y2x

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V. NEW! MULTIPLICATION OF POLYNOMIAL EXPRESSIONS

A Multiplication of a monomial and a binomial

We will build on the knowledge we have reviewed and multiply a monomial times a

binomial and a monomial times a trinomial. Lets begin with an example of multiplication of a

monomial times a binomial.

To multiply a binomial (i.e., two term polynomial) by a monomial, you multiply each term of

the binomial by the monomial. This process uses the distributive property.

Example 1: 2(3x 1)

= (2)(3x) + (2)(-1)

= 6x 2

Example 2: 6x(12x + 6) = 6x(12x) + 6x(6)

= 72x3 + 36x

Example 3: -2y(2x 4y) = -4xy + 8y

B Multiplication of a monomial and a trinomial

The algebraic method and distributive property can be used to multiply monomials by

trinomials as well as the multiplication of more complex expressions. Carefully read the

examples below.

Example 1 2x(3x2 4x + 1)

= (2x)(3x2) + (2x)(-4x) + (2x)(1)

= 6x3 8x2 + 2x

We must multiply the 2 by eachterm of the binomial. The

multiplication distributes overthesubtraction.

We must multiply the 6x by eachterm of the binomial. Themultiplication distributes overthesubtraction.

Each term inside the ( ) ismultiplied by 2x. Multiplication isdistributed throughout the ( ).

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Example2 3x2y(4x2y 2x + 3y)

= (3x2y)(4x2y) + (3x2y)(-2x) + (3x2y)(3y)

= 12x4y2 6x3y + 9x2y2

In these two examples we multiplied a monomial times a trinomial. You can see theprocess does not change as we add terms.

Hippo Campus Do lessons 2 and 3 listed in the menu on the left at this link.

To practice the concept, click on the word homework under the lesson title. If

you have trouble using this website a video is provided in D2L under

contentInstructional videos for websites.

Look under content in D2L to view the In a nutshell online seminar called

Multiplication of Polynomials using the Distributive Property.

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C MUTIPLYING TWO BINOMIALS

To multiply two binomials, we use the distributive property of multiplication over addition.

This is sometimes referred to as the FOILMethod [FIRST, OUTER, INNER, LAST]. Remember to

combine any like terms after multiplying.

Example 1: (6x + 2)(4x 5)

Solution: (6x + 2)(4x 5) = 6x(4x) + 6x(-5) + 2(4x) + 2(-5)

= 24x 30x + 8x 10

= 24x 22x 10

Example 2: (2a + 3b)(4c 2d)

Solution: (2a + 3b)(4c 2d) = 2a(4c) + 2a(-2d) + 3b(4c) + 3b(-2d)

= 8ac 4ad + 12bc 6bdLook closely at the next 3 examples to see if you can see a pattern that will allow you to develop

shortcuts for each.

Example 3: (2x + 3)

Solution:

(2x + 3) = (2x + 3)(2x + 3)

= 4x + 6x + 6x + 9

= 4x + 12x + 9

Example 4: (2x - 3)

Solution:

(2x - 3) = (2x - 3)(2x - 3)

= 4x - 6x - 6x + 9

= 4x - 12x + 9

Example 5: (2x + 3)(2x-3)

Solution:

=(2x + 3)(2x-3)

= 4x - 6x + 6x - 9

= 4x - 9

Follow the link to see what makes examples 3, 4, and 5 special.

Hippo Campus Do lesson 4 to see more examples of multiplication. Then,do lesson 5 to discover patterns and shortcuts when multiplying twobinomials. To practice these concepts, click on the word homework underthe lesson title. If you have trouble using this website a video is providedin D2L under contentInstructional videos for websites.

Look under content in D2L to view the In a nutshell online seminar calledFOIL.

Try the Homework Assignment called Multiplying two Binomials FOIL -LG page 22 to insure you can use the FOIL method.

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Do Exercise 5 in your notebook. Check your answers before moving onto the next section.

EXERCISE 5

Complete the following algebraically using the distributive property,

1. (x 2)(x + 5) 4. (4s+3)2

2. (2x 3)(x 2) 5. (2k-5)2

3. (-x + 1)(x 4) 6. (w+6)(w-6)

ANSWERS TO PRACTICE EXERCISE 5

1. (x 2)(x + 5) 4. (4s+3)2

= (4s+3)(4s+3)

= (x)(x) + (x)(5) + (-2)(x) + (-2)(5) =(4s)(4s) +(4s)(3) +(4s)(3) + (3)(3)

= x2

+ 5x 2x 10 = 16s2

+ 12s +12s + 9

= x2

+ 3x 10 = 16s2

+ 24s + 9

2. (2x 3)(x 2) 5. (2k-5)2

= (2k-5)(2k-5)

= (2x)(x) + (2x)(-2) + (-3)(x) + (-3)(-2) =(2k)(2k) +(2k)(-5) +(2k)(-5) + (-5)(-5)= 2x

2 4x 3x + 6 = 4k

2-10k -10k+25

= 2x2 7x + 6 = 4k

2-20k +25

3. (-x + 1)(x 4) 6. (w+6)(w-6)

= (-x)(x) + (-x)(-4) + (1)(x) + (1)(-4) = (w)(w) + (-6)(w) + (6)(w) + (-6)(6)

= -x2

+ 4x + 1x 4 = w2- 6w + 6w -36

= -x2

+ 5x 4 = w2 36

Try the Homework Assignment called Polynomials FOIL Special CasesLG page 23 to insure you can use the FOIL method for any pair ofbinomials.

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Using the Distributive Property to Multiply two Polynomials

The distributive property of multiplication over addition can be applied when multiplying a monomial

by a trinomial, a binomial by a trinomial, etc.

Example 1: 2a(3a + 4b + 2c) = 6a + 8ab + 4ac

Example 2: (x 1)(x + 2x 1) = x(x) + x(2x) + x(-1) 1(x) 1(2x) 1(-1)

= x3 + 2x x x 2x + 1

= x3 + x 3x + 1

Example 3: ( )( ) ( )

( )

( )

Use your ebook to see 2 interactive examples on Page 185. Theseinteractive activities can also be accessed using the Media tab andselecting Lesson 3.7 Multiplying Polynomials Exercise 3a andexercise 3b.

Look under content in D2L to view the In a nutshell online seminarcalled Using the Distributive Property to Multiply any two Polynomials.

Read The examples on pages 161, 170, 183-184 of your text. Use the

distributive property (FOIL) method to practice with the Check YourUnderstanding exercises on these pages.

For additional practice try some/all of the following:page 166 167 #s 5, 9, 12, 18page 177 #s 8, 9, 10page using the FOIL method. You do not have to sketch the algebratiles.Page 184 #s 6 , 7, 10, 12, 19 (all questions from 4-21 are good practice)

collect like

terms

NOTE: It is easier tomultiply the binomialsBEFORE multiplying by aconstant.

RememberBEDMAS
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Try the Homework Assignment called Multiplying Bi and Trinomials LGpage 25 to insure you can use the FOIL method for any pair of binomials.

Do Exercise 6 in your notebook. Check your answers before moving onto the next section.

EXERCISE 6

Simplify the following expressions.

1. 3x(4x3y)(-2x

3y) 10. (x y)(x y)

2. (-5x)(6y)(-2z) 11. 5(x

3x)

2(x

8)3. 6x(3x 2y) 12. -3(2x + x) + 6x(2 + 3x)

4. 2(12abc 13cd) 13. (2x 1)(x + 2) + (4x + 1)(x + 1)

5. -2e(4f + 6g6h) 14. -6(x 2) 4(x 1)(x + 4)

6. (x + 5)(x 3) 15. (x 4) + (x + 1)

7. (x + 4y)(x 3y) 16. (2x 1) (2x + 1)

8. (a + b)(ab 2ab + ab) 17. (x 3)(x 2) + 2(x + 4)

9. (a b + c)(a + b c) 18. a(a + b + c) 2b(a + b c)

SOLUTIONS EXERCISE 6

1. -24x7y3 11. 5x 17x + 16

2. 60xy 12. 12x + 9x

3. 18x 12xy 13. 6x + 8x 1

4. 24abc 26cd 14. -4x 18x + 28

5. -8ef 12eg + 12eh 15. 2x 6x + 17

6. x + 2x 15 16. -8x

7. x + xy 12y 17. 3x + 11x + 38

8. a3b 2ab + 2ab 2ab + ab

318. a 2b ab + ac + 2bc

9. a b c + 2bc

10. x 2xy + y

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The Process of Verification

It is often important that we know how we can check our work. The multiplication of

polynomials should be checked or verified. To verify a product, we can substitute a number in

for the variable in both the original question and our product. If we have multiplied correctly,

then the two values will be the same. Carefully read the example below to see how this is done.

Example (2x + 3)(x 4)

= (2x)(x) + (2x)(-4) + (3)(x) + (3)(-4)

= 2x2 8x + 3x 12

= 2x2 5x 12

Verification Let x = 1

(2x + 3)(x 4) = 2x2 5x 12

(2(1) + 3)((1) 4) 2(1)2 5(1) 12

(5)(-3) 2 5 12

-15 -15

Since both the original and the product equal -15, we know that

(2x + 3)(x 4) = 2x2 5x 12.

Example 2 (3x 4)(2x + 5)= (3x)(2x) + (3x)(5) + (-4)(2x) + (-4)(5)

= 6x2 + 15x 8x 20

= 6x2 + 7x 20

To verify, let x = 2 (3x 4)(2x + 5) = 6x2 + 7x 20

(3(2) 4)(2(2) + 5) 6(2)2 + 7(2) 20

(6 4)(4 + 5) 6(4) + 14 20

(2)(9) 24 6

18 18

Since both sides equal 18, (3x 4)(2x + 5) = 6x2 + 7x 20

N.B. We can use anyvalue in our check!

N.B. Notice the format of a verification! There isa ? over the equal sign and a line separating thetwo sides of the equation. This is necessary untilwe have shown the two sides to be equal!

?

?

Notice order ofoperations! Wesquare 2 then multiplyby 6.

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The algebraic method is most widely used and will be the most efficient method when

determining more complex products such as the one that follows below.

Example (3x

2

+ 2)(x

2

2x)= (3x2)(x2) + (3x2)(-2x) + (2)(x2) + (2)(-2x)

= 3x4 6x3 + 2x2 4x

We can use algebra and the distributive property to multiply any number of terms by any

number of terms.

Lets show this by multiplying a trinomial times a trinomial as in the example that follows.

Example (x2 + 2x + 3)(2x2 3x + 1)

First term in first trinomial is multiplied by each term in second trinomial

= (x2)(2x2) + (x2)(-3x) + (x2)(1)

= 2x4 3x3 + x2 result 1

Second term in first trinomial is multiplied by each term in second trinomial

= (2x)(2x2) + (2x)(-3x) + (2x)(1)

= 4x3 6x2 + 2x result 2

Third term in first trinomial is multiplied by each term in second trinomial

= (3)(2x2) + (3)(-3x) + (3)(1)

= 6x2 9x + 3 result 3

The final product is result 1 + result 2 + result 3.

2x4 3x3 + x2 + 4x3 6x2 + 2x + 6x2 9x + 3

= 2x4 3x3 + 4x3 + x2 6x2 + 6x2 + 2x 9x + 3

= 2x4 + x3 + x2 7x + 3

When multiplying polynomials, it is important to remember that each term in

the first polynomial is multiplied by each term in the second polynomial.

Simplify the expression bycombining like terms.

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Lets verify using x = -1 It is necessary to place -1 in parentheses as shown below!

(x2 + 2x + 3)(2x2 3x + 1) = 2x4 + x3 + x2 7x + 3

((-1)

2

+ 2(-1) + 3)(2(-1)

2

3(-1) + 1) 2(-1)

4

+ (-1)

3

+ (-1)

2

7(-1) + 3(1 + -2 + 3)(2 + 3 + 1) 2 + -1 + 1 + 7 + 3

(2)(6) 13 + -1

12 12

Since both sides worked out to be 12,

(x2 + 2x + 3)(2x2 3x + 1) = 2x4 + x3 + x2 7x + 3

Now that you understand how to multiply polynomials, you will be asked to demonstrate

your understanding in a variety of ways. Carefully read the following examples.

Example 1 Suzies solution to a multiplication question appears below. Identify and explainthe error that she made.

(3x 2)(2x + 5)

= (3x)(2x) + (-2)(5)

= 6x2 10

Example 2 Explain to Justin the error that he made when he multiplied the following

binomials.

(4x 1)(2x 3)

= (4x)(2x) + (4x)(-3)

= 8x2 12x

Refer to Foundations and Pre-Calculus Mathematics

10. On page 167, complete Question 13

To summarize, when multiplying polynomials it is necessary to

Multiply each term in the first polynomial by each term in the second polynomial.

Simplify the product by combiningliketerms using addition or subtraction.

?

Suzie multiplied only the first term ineach bracket and the last term in eachbracket. She needs to multiply eachterm in the first binomial by each term inthe second binomial.

Justin multiplied only the first term,4x, in the first binomial with eachterm in the second binomial. Heneeds to multiply each term in thesecond binomial by -1, the secondterm in the first binomial.

Note!

(-1)2 = 1

(-1)3 = -1
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POLYNOMIALS

PRACTICE TEST I

PART A

For each polynomial, determine the number of terms, name the polynomial, give the degree of

each term and give the degree of the polynomial.

1. 4x2y3 + x4

2. 8x2y4 + 7x2y5 2x6 + 1

PART BSimplify by performing the operation indicated.

1. 7xy2 2x2y + 3xy 5x2y

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

PART C

1. Use algebra to multiply the following polynomials.

(a) -3x(4x 3)

(b) (3x 5)(2x 7)

(c) (2x + 3)(3x2 4x 1)

(d) 2

2 1x

(e) 4(2x 3y)(2x + 3y) (2x + 5y)(3x y)

(f) ( 3)(2 1)( 4) x x x

PART D

Determine the error in the following work, correct the error and complete the question. Verify

your new solution.

-(3x 2)(2x + 3)

= -(6x2 + 9x 4x 6) Step 1

= -6x2 + 9x 4x 6 Step 2

= -6x2 + 5x 6 Step 3

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PART E

Each shape is a rectangle. Write a polynomial to represent the area of the shaded region.

Solutions to the Practice Test appear on the following pages.

Try the Quiz P1-2 Checkpoint 1 to test yourself on this unit.

3 + 8

2 + 6

+ 6

3

x

x

x

x

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SOLUTIONS TO PRACTICE TEST

PART A

PART B

1. 7xy2 7x2y + 3xy

2. -4x2 + 4

PART C

1. (a) -12x2 + 9x

(b) 6x2 31x + 35

(c) 6x3 + x2 14x 3

(d) 24 4 1x x

(e) 4(2x 3y)(2x + 3y) (2x + 5y)(3x y)

= 4(4x2 9y2) (6x2 2xy + 15xy 5y2)

= 16x2 36y2 6x2 13xy + 5y2

= 10x2 13xy 31y2

(f)

2

3 2 2

3 2

3 (2 1)( 4)

(2 7 3)( 4)

(2 8 7 28 3 12)

2 25 12

x x x

x x x

x x x x x

x x x

Numberof terms

Name ofpolynomial

Degree ofeach term

Degree ofpolynomial

1. 4x2y3 + x4 2 Binomial 5, 4 5

2. 8x2y4 + 7x2y5 2x6 + 1 4 Polynomial 6, 7, 6, 0 7

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APPENDIX

. (Go to grades 7-9). Look up the word polynomial.

A polynomial is an expression which is a monomial, or the sum (or difference) of two or more

monomials.

Now look up the word monomial.

A monomial is a term consisting of a number, a variable, or a product of numbers and variables.

Now look up the word term

A term is any individual element of a mathematical expression.

Often a term refers to a monomial (a number, a variable, or a product of numbers and

variables).

Connect the key words in these definitions.

A polynomial is made up of monomials.

monomials are terms.

terms are a number, a variable, or the products of numbers and variables

Therefore; any algebraic expression that contains an element (term) that uses an operationother than multiplication is not a polynomial.

Division by a variable was used in the examples of algebraic expressions that are notpolynomials. Therefore they are not polynomials.

Example: In the algebraic expression

the variable is the divisor. In R1-3 you have, or

will learn that such expressions can be written in exponent form as . Therefore anyalgebraic expression that has a negative exponent or a variable as a divisor is not a polynomial.

You will study this concept in more detail in R1-3 and in your grade 11 math course.
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