MathChapter 1s1

8/3/2019 MathChapter 1s1

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Mathematic for Junior High School Year 8 1

In this chapter, you will learn polynomials, and how to do

additions, subtractions, and multiplications on algebraic expressions.

A number of concepts that you will learn are some types of

polynomials, which include monomial and binomial. You have to

understand how to do operations (additions, subtractions, or

multiplications) on polynomials and how to simplify some

polynomials by using these operations.

In dealing with algebraic concepts and these related topics,

polynomials are important concepts that you have to understand.

These concepts are widely used not only in algebraic concepts but also

in any other concepts which use polynomials as expressions for

representation.

The key terms that you need to consider are polynomials,

factors, factorizations, GCD (Great Common Divisors), perfect

quadratic expressions, and algebraic expressions.

The following diagram shows the map of polynomials, and

factorization of algebraic expressions.

Powers(Exponents)

Polynomials

Terms VariablesCoefficientss


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Students Book 8 Polynomials2

The Meaning of a Polynomial

Consider the following algebraic expression:

5a3 + 4a2a2 + 9a + 6

This algebraic expression is called a polynomial. It

has 5 terms, namely 5a3

, 4a2

, a2

, 9a and 6. In 5a3,5 is called the coefficient of a3, 3 is called the poweror

exponent of a, and a is called a variable. Similarly, in

4a2, 4 is called the coefficient of a2 and 2 is called the

power or exponent of a.

Some polynomials have specific names. A

polynomial that has only one term is called a

monomial, the one having two terms is called a

binomial, and the one having three terms is called

trinomial. Of course, a special name can also be

given to a polynomial having more than three

terms. In general, a polynomial is either a

monomial or a sum of monomials. Note that a

constant can be considered as a monomial.

The following is an example of the application of polynomials in our real

life. Daddy went to a fast-food restaurant. Daddy ordered 2 packs of French

fries, 3 pieces of hamburgers, and 1 glass of soda. If the prices of one pack of

French fries, one piece of hamburger, and one glass of soda are denoted by f, h,

and s, respectively, then the total price would be:

2f+ 3h + s.

Learning Objectives:to identify polynomials

to simplify polynomials

by grouping like terms

to determine themultiplication of a

monomial and a

binomial

to determine themultiplication of twobinomials

to determine the

multiplication ofbinomials

to determine the

square of binomials

Key Terms:

like terms polynomial monomial binomial trinomial distributive property


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This expression is a polynomial, specifically it is a trinomial. If, in fact,

f= Rp 10,000, h = Rp 12,000, and s = Rp 5,000, then the total price can be calculated

as follows:

The total price = 2(Rp 10,000) + 3(Rp 12,000) + 1 (Rp 5,000)

= Rp 20,000 + Rp 36,000 + Rp 5,000

= Rp 61,000

The following are some other examples of polynomials:

Polynomials Examples

Monomial a. 2x2b. 5cc. 10

Binomial a. 5h + 2fb. 8c + 2c. c2 + 3c

Trinomial a. 3h + 2f+ mb. 5w2 + 36w + 4c. c2 - 5c + 2

More than 3 terms a. 2x3 + 4x2 - x 7b. 2x5 + 3x4 -5x3 + x2 x - 7

A polynomial is quite often written in a descending order. This means that

the polynomial is started with the term having the largest power followed by the

lower one.

Simplifying Polynomials

Look at the following polynomial:

5a3 + 4a2a2 + 9a + 6.

In this polynomial the terms 4a2 and a2 are like terms, the terms having the same

variables of the same power. A polynomial having like terms can be simplified by

adding or subtracting the like terms. So, for example, the polynomial 5a3 + 4a2a2 + 9a

+ 6 can be simplified as follows:


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5a3 + 4a2a2 + 9a + 6 = 5a3 + (4-1) a2 + 9a + 6

= 5a3 + 3a2+ 9a + 6

The last expression is the simplest one that consists of 4 terms, that is , 5a3, 3a2, 9a and 6.

Below are some other examples of polynomials.

Polynomial:2c + s +f+ s + h + c - s

Grouping the terms:( 2c + c ) + ( s + s - s ) +f+ h

The simplest expression:3c + f + h + s

Polynomial:n + x + y2+ 2x + y2

Grouping the terms:( y2 + y2 ) + ( x + 2x ) + n

The simplest expression:

2y2 + 3x + n

Write each of the following polynomials in the simplest expressions.

a. 4x 2x b. 5 + 2x 1 c. 3x 6x + 4

d. 8 + 3xx 6 e. 6 + 6x f. 3x + 3xxg. 4x2x h. 5x2 + 2x 3 i. 2x3 3xx2 + 2x + 5

Points to remember

Terms of polynomials can be a number, a variable or a product of a number

and a variable.

The coefficient of a variable is a number that is a multiplier of a variable.

Like terms are the terms which have the same variables of the same power.

A constant term is a term that consists of a number only.


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1. Simplify the following polynomials.a. 2n 3n b. x + 7 + 3x

c. 2k 5bb k d. 7y2 3y + 4y + 8y2 + 4y

e. 2x2 4 + 3x2 6 x2 f. c2 + 2cc2c

g. 2 + 6x + z 2x + 8 4z h. 9p + 10 p + 3x 5

i. 4 + k 9mm + 2k

2. Write down one example of a monomial, one example of a binomial, and one

example of a trinomial. Explain.

3. Write down a polynomial containing four terms that can be simplified into

binomials.

4. The sizes of two angles of a triangle areshown below. Find the sizes of the angles.

(2x 2)

(x + 10)

Summary

1. A polynomial is either a monomial or a sum of monomials. A monomial

can be a number, a variable, or multiplication of a number and avariable.

2. A polynomial containing two terms is called a binomial, whereas apolynomial containing three terms is called a trinomial.

3. To simplify a polynomial, we should group like terms, and then calculatethem.


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Multiplication of a Monomial and a Binomial

In this section you will learn about the multiplication of a monomial and a

binomial. Consider the following situation: Mr. Harso asks Andi to calculate the area of a rectangle whose length is 2

centimetres longer than its width.

How do you solve this question?

Suppose the width of the rectangle is w cm. Then its length is l = (2 + w)cm.

Thus, the area of the rectangle is A = l w cm2 = (2 + l)l cm2. This expression is

an example of multiplication of a monomial and a binomial.

Now let us do the following Mini Lab activity.

MULTIPLICATION OF A MONOMIAL AND

A BINOMIAL

Group ActivityMaterial: tile model

A tile model is constructed based on the

area of a square or rectangle. The area of a

rectangle is the product of its length and its width.

You can use a tile model to have more complex rectangles. These

rectangles will help you to understand how to determine the

product of simple binomials.

The length and width of the rectangle are the factors being

multiplied.

Your task

Work with your classmates to determine x(x + 2).

Use the following hints.

Draw a rectangle with the length of (x + 2) and the width ofx.

Use the tile model to identify the factors.

Use those factors as a guide to fit the rectangle into the tile

model.

MINI - LAB

1xx

1

x2

x

x

1

1


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In the Mini Lab activity, we have determined the area of a rectangle using

polynomials. Now, we will use the distributive property that you learned

previously in Year 7.

The product of a monomial and a binomial can be modeled as the area of arectangle which is formed using the tile model.

Polynomial (x + 2)2x can be modeled as the area of a rectangle of x + 2 inlength and 2x in width.

The result of (x + 2)(2x) can be determined in two ways:Method I:Add the areas of the tile model.

x2 +x2 + x + x + x +x = 2x2 + 4x 2x

x + 2

x2 x x

x2 x x

Determine the area of the rectangle in two ways:

Method I : add the area of the tile model

Method II : use the formula of the area of a rectangle and use the

distributive law of multiplication over summation.

Compare the answers.

Discuss the following problems

1. Say whether each of the following statements is correct or wrong. Check

your answer using the tile model.

a. x(2x + 3) = 2x2 + 3x b. 2x(3x + 4) = 6x2 + 4x

2. Determine the result of each of the following multiplications using the tile

model.

a. x(x + 5) b. 2x(x + 2) c. 3x(2x + 1)

3. Suppose Agus has a square garden with the

sides ofx metres in length. If Agus wants to

enlarge his garden so that its length is twice

the length of the previous one and its width

is 3 metres more than the previous one, what

is the area of Aguss new garden?

MINI - LAB

x

x 1 1

x2 x x


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Method II:Apply the distributive law:

(x + 2)(2x) = (x)(2x) + (2)(2x)= 2x2 + 4x

Expand the product of the following multiplication of a monomial and a

binomial using the distributive law.

a. 7(2x + 5) b. (3x7) 4x

Multiplication of Two Binomials

To understand the multiplication of two binomials,

consider the following situation. Suppose you have

a rectangular garden. The length of the garden is

five metres longer than twice of its width. On the

periphery of the garden there is a road of 1 metre

width as shown in the figure. The area of the road is24 m2. What are the length and width of the garden?

To answer the question, you can use a tile model.

Suppose that x represents the width of the garden. Then

2x + 5 represents the length of the garden.

x + 1 represents the width of the garden and the road.

2x + 6 represents the length of the garden and the road. Thus x(2x + 5) equals the area of the garden.

(x + 1)(2x + 6) equals the area of the garden and the road.


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2x + 6

2x + 5

x + 1 x

Plan:

(x + 1)(2x + 6) x(2x + 5) = 24 (*)

Solution: (x + 1)(2x + 6) x(2x + 5) = 24

2x2

+6x + 2x + 6 2x2

5x = 24(2x2 2x2) + (6x + 2x 5x) + 6 = 24

3x + 6 = 24

3x = 18

x = 6

Hence, the width of the garden is 6 m.

The length of the garden is 2x + 5 = 2(6) + 5 = 17 meters

Check whether the result is correct if x = 6 is substituted into the equation

(*).

Expand (x + 3)(x + 2) by referring to the Mini Lab activity on page 6.

Explain the steps that you use.

1. Expand the following multiplications and explain the steps you use.

a. (2x + 3)(3x + 5) b. (2x + 1)(5x 3)

2. Fika expands the multiplication of two binomials using a method that is

called FOIL (First, Outer, Inner, Last). Fika explains the FOIL method asfollows.

Area of garden

and streetArea of

garden Area of street


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F: means Fika multiplies the first term of the first binomial with thefirst term of the second binomial.

O: means Fika multiplies the outer terms, i.e. the first term of the first

binomial and the second term of the second binomial.I: means Fika multiplies the inner terms, i.e. the second term of thefirst binomial and the first term of the second binomial.

L: means Fika multiplies the last (second) terms of the binomials.

a. Use the Fikas method to expand (3x + 5)(2x + 7).

b. Expand (3x + 5)(2x + 7) using the ways you have learned (usingalgebraic tiles as a model or using the distributive property).

c. Compare your results in part (a) and part (b).

3. Expand the following multiplications using the Fikas method.a. (x + 3) (x + 5) b. (2y + 3) (3y + 4)c. (a 1) (a 7) d. (6x + 1) (2x 3)

Summary:Multiplication of two binomials can be modeled by using algebraic tiles.

An algebraic expression such as (x + 2)(2x + 3) can be modeled as a

rectangle having the length of (x + 2) and the width of (2x + 3).

The expression (x + 2)(2x + 3) represents the area of the rectangle. To

expand this expression you can use algebraic tiles as a model, use the

distributive property, or use the FOIL (First, Outer, Inner, and Last)

method.

The expansion of (a + b)(c + d) by using FOIL method is illustrated as

follows.

(a + b) (c + d)


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Powers of Monomials and Binomials

In elementary school, you have learned about the power of an integer.

You should be able to answer the following questions.

What does 73 mean?

How to determine the value of 7 3? What is the value?

If k is a number, what does k4 mean?

The expressions 73 and k4 can be considered as powers of polynomials

(monomials).

Discussion

Suppose Mr. Budi has a square garden with the length of (x + 5).

a. Express the area of Mr Budis garden as a function of x.

b.Is the area of Mr Budis garden a power of a polynomial?

How do you calculate the result of (x 2)3?


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1. Explain how to calculatethe multiplication of x and 2 x 1.

2. Explain why x(2x + 3) and (2x + 3)x are equivalent.

3. Use algebraic tiles to calculate each of the following multiplications.

a. 5(x + 2) b. x(x + 4) c. 2x(x 1)

4. Determine the results of the following multiplications.

a. 7(3x + 5) b. 2(x + 8) c. y(y 9)

d. pq(pq + 8) e. 7(2a2 + 5a11) f. 3y(6 9y + 4y2)

g. 2(n 6) h. (5b 4) 52

i. 52

(5w + 10)

5. Simplify each of the following algebraic expressions.

a. 18y + 5(7 + 3y) b. 14(b + 3) + 8b c. 30(b + 2) + 2b

d. 3(8 + a) + 7(6 + 4a) e. x + 5x + 8(x + 2) f. 3(x + y) + 4(2x + 3y)

6. Calculate the area of the shaded region

on the figure on the right.

7. Is 2ab = 2a 2b? Explain your answer. 8. Explain the similarities between the

procedure of multiplication of two binomials and the procedure of

multiplication of a monomial and a binomial. Explain the differences.

9. Draw a rectangle having an area of (x + 3)(2x + 1).

10.Use algebraic tiles to calculate the following multiplications.

a. (x + 1)(x + 2) b. (x + 3)(x + 4) c. (2x + 3)(x + 2)

11.For each of the following models, show the two binomials being multipliedand then write down the results.

a. b.x2 x2 x x

x x 1 1x x 1 1

x2 x x x

x 1 1 1

x2 x x x

2s s

s

3


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12.Calculate the result of each of the following multiplications and use the

FOIL method.

a. (x + 2)(x + 2) b. (x 6)(x + 2) c. (x + 7)(x 5)

d. (2x + 3)(x 4) e. (3x 4

1

)(6x 2

1

) f. (x 2)(x2

+ 2x)

13. Simplify the following expressions.

a. (p 3)2 b. (3+ 2t)2 c. (2x 1)2

d. (x 4)3 e. 3a2 + (2a + 1)2 f. (x 1)3 + (x + 7)2

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MathChapter 1s1