Using Algebraic Tiles on the Factorisation

Lia wants to frame a

Work in pairs to help Lia find the length and the width.

a. Why cannot 221 be expressed as a product of two one-digit numbers?

b. Use a piece of grid paper and do the following

steps:

- Cut some squares having the size of 10×10. - Cut some rectangles having the size of 1×10. - Cut some squares having the size of 1×1.

- Use the pieces of paper to make a rectangle

Using Algebraic Tiles on the Factorisation

Factorising a number means representing the number as a multiplication of some other numbers. For example, what are the factors of 12? The number 12 can be factorised as follows.

12 = 1 x 12 12 = 3 x 4 12 = 3 x 2 x 2 12 = 6 x 2

On the factorisation 12 = 1 x 12, 1 and 12 are the factors of 12. As on 12 = 3 x 4, 3 and 4 are the factors of 12.

Consider the following algebraic form

2x(2y + 3),

the expression 2x and (2y + 3) are the factors.

The next problem is to factorise using algebraic tiles. For that purpose, do the following Mini Lab activity first.

FACTORIZATION

Group Activities

Material: Algebraic Tiles

Suppose a rectangle has the length of (x + 3) and the width of (x + 1). Then (x + 1)(x + 3) = x2 _{+ 4x + 3.}

In other words, the factors of x2 _{+ 4x + 3 are (x + 1) and (x + 3).}

Your task:

Work in group of three to factorise x2 + 3x + 2.

¾ Make a model of the trinomial.

x2 _{x x x}

x 1 1 1

x2 _{x x x}

Mini – Lab

¾ Arrange x2 and 1 tiles as shown beside.

¾ Put x tiles to form a rectangle.

Because a rectangle can be formed, x2+ 3x + 2 can be factorized. The length of the rectangle is (x + 2) and its width is (x + 1). Thus, the factors of x2+ 3x + 2 are (x + 1) and (x + 2).

factorised. The length of

PROBLEM 1

1. Determine whether each of the following polynomials can be factorised. Check your answer using Algebraic Tiles.

a. x2 _{+ 6x + 8 b. x}2 _{+ 5x + 6}

c. x2 _{+ 7x + 3 d. 3x}2 _{+ 8x + 5}

e. 5x2 _{– x + 16 f. 8x}2 _{– 31x – 4}

2. Find examples of a trinomial that can be factorised and a trinomial that cannot be factorised.

Checking the Possible Factors

Look at how Rahmah factorises x2 _{+ 7x + 12 by doing “Trial and error” and} uses Algebraic Tiles.

First, Rahmah takes one tile of x2_{, seven tiles of x, and twelve tiles of 1.}

She, then, constructs a rectangle using all of the tiles.

x + 1

Experiment I Experiment II Experiment III

The method Sahrul used to factorise a trinomial can be illustrated as follows.

x

2

_{+ b}

x

_{+ c = (}

x

₊

†

_{) (}

x

₊

†

₎

The sum o f the se num b e rs e q ua ls b

The p ro d uc t o f the se num b e rs e q ua ls c

PROBLEM 2

Use the above schema to find the factors of x2 _{– 2x – 8. Draw a table as what} Sahrul did.

Factorisation by Separating GCD

Ani illustrates how to factorise 2x2 _{– 10x using Algebraic Tiles.}

x

2x

x x

x – 5 -1 -1 -1 -1 -1

This model shows that 2x2 _{– 10x = 2x (x – 5).} Mo d e l o f 2x2 _{– 10x}

A re c ta ng le is fo rm e d using the m o d e ls.

1

2

3

The GCD (Greatest Common Divisor) of 2x2 _{and 10x is 2x. Using the} distribution rule, it can be concluded that:

2x2 _{– 10x = 2x(x) – 2x(5) = 2x(x – 5)}

Thus, factorisation can also be done, first by separating its GCD and then applying the distributive rule.

To clarify this method, consider the following example.

EXAMPLE 3

Then, apply the distributive rule for separating the common divisor.

3x3 _{– 9x}2 _{+ 15x = 3x(x}2_{) – 3x(3x) + 3x(5)} = 3x(x2 _{– 3x + 5)}

Factorising ax

2

+ bx + c

,

when a

≠

1

3. Use the factors to form binomial.

4. Place the factors of 3 inside the box

†

,

and the factors of –6 inside the circle c in the following expression

(

†

x

₊

c

₎₍

†

x

₊

c

₎

_.

5. Find the multiplication of two binomials. Its middle term, which is the sum of outer and inner, is equal to –7x.

( 1x + 1 ) ( 3x + –6 ) –6x + 3x = 3x WRO NG

( 1x + –6 ) ( 3x + 1 ) 1x – 18x = –17x WRO NG

( 1x + –1 ) ( 3x + 6 ) 6x – 3x = 3x WRO NG

( 1x + 6 ) ( 3x + –1 ) –1x + 18x = 17x WRO NG

( 1x + 2 ) ( 3x + –3 ) –3x + 6x = 3x WRO NG

( 1x + –3 ) ( 3x + 2 ) 2x – 9x = –7x C O RREC T

3

Using the above method, factorise 6x2 _{– x – 2.}

inner

outer

SUMMA RY

1. The fa c to risa tio n o f a p o lyno m ia l c a n b e d o ne using Alg e b ra ic Tile s. If a re c ta ng le c a n b e fo rm e d using a ll units, the p o lyno m ia l c a n b e fa c to rise d . The fa c to rs a re the le ng th a nd the w id th o f the re c ta ng ula r. O n the o the r ha nd , if the re c ta ng ula r c a nno t b e fo rm e d , the n the p o lyno m ia l c a nno t b e fa c to rise d .

2. Ano the r w a y to fa c to rise is b y se p a ra ting the G C D o f the p o lyno m ia l te rm s, a nd the n a p p lying the d istrib utive rule .

3. Fa c to risa tio n o f p o lyno m ia ls c a n a lso b e d o ne b y a p p lying g ue ssing a nd c he c king stra te g y.

Factorising the Difference of two Squares

Work in pairs. Find the polynomials of the following factors in Groups A, B, and C of the following table. Then, answer the questions.

Group A Group B Group C

(x + 7)(x – 7) (x + 7)(x + 7) (x – 7)(x – 7) (k + 3)(k – 3) (k + 3)(k + 3) (k – 3)(k – 3) (w + 5)(w – 5) (w + 5)(w + 5) (w – 5)(w – 5) (3x + 1)(3x – 1) (3x + 1)(3x + 1) (3x – 1)(3x – 1)

1. What is the pattern of binomials pairs in each group? 2. What are the products?

3. Check the products in each group.

As we can see in the “Group Activity” above, sometimes when we multiply two binomials, the middle term of the product is 0.

4. What polynomial can be represented by each group of the

following models? Find the factors of each polynomial.

x2 x x

5. Examine whether the polynomial resulting from answer No. 4 and

the ones from Group A above can be written as the difference of two squares or a2 _{– b}2_.

6. Conclude how to factorise the difference of two squares.

7. By using your conclusion above, factorise the following algebraic forms.

8. a. Write down at least two binomials of the 7a type. What are the characteristics of the binomials?

b. Write down at least two binomials of the 7b type. What are the characteristics of the binomials?

9. Factorise the following binomials orally.

a. x2 _{– 36 b. 9y}2 _{– 25}

10.

Critical Thinking

Suppose one of your friends tries to factorise 4x2 _{– 121 and gets the result of (4x + 11)(4x – 11). What}

mistakes has he made? Explain your answer.

Factorising Perfect Quadratic Trinomials

In the last “Group Activity”, you multiplied a binomial by itself as in Groups B and C. These multiplications are called squaring binomials. The result is called a perfect quadratic trinomial. Therefore, the factors of a perfect quadratic trinomial are two identical binomials.

Discuss

How do you know that a trinomial is a perfect quadratic form?

Use the FOIL method to determine the multiplication results of the following binomials.

a. (a + b)(a + b) b. (a – b)(a – b)

The results of the above multiplications are called perfect quadratic trinomials.

For all real numbers a and b,

a2 _{– b}2 _{= (a + b)(a – b)}

The Difference

of Two

Use Algebraic Tiles to show how to factorise the following perfect quadratic form trinomials. What is the factorisation pattern?

Don't

forget

Check the factors you get by multiplying them back.

a. x2 _{+ 8x + 16 b. x}2 − _{8x + 16}

a. Write down another trinomial which is a perfect quadratic

form.

b. Explain how you know that a trinomial is a perfect quadratic trinomial.

Sometimes, it seems that a quadratic form cannot be factorised. If you find this case, first separate the common factors, and then, check if there is any of the remaining factors that can be refactorised.

Factorise 10x2 _{– 40.}

Solution:

10x2 _{– 40 = 10(x}2 _{– 4)}

= 10(x + 2)(x – 2)

Common divisor of 10x2 _{and 40}

is 10

Factorise x2_{– 4}

Thus 10x2 _{– 40 = 10(x + 2)(x – 2).}

For each real number a and b,

a2 _{+ 2ab +b}2 _{= (a + b) (a + b) = (a + b)}2

a2 _{– 2ab +b}2 _{= (a – b) (a – b) = (a – b)}2

Examples:

x2 _{+ 10x + 25 = x}2 _{+ 2.5.x + 25 = (x + 5)(x + 5) = (x + 5)}2

Perfect

Quadratic

Trinomials

1. Write down the length and the width of each of the following rectangles as binomials. Then write down the algebraic form for each of the following rectangles.

a. b. c.

2. Can you construct a rectangle using all given tiles? Explain your answer. a. One x2_{, two x, and one unit 1.}

b. One x2_{, five x, and eight 1.}

3. Determine the GCD of terms of each of the following polynomial.

a. 15x + 21 b. 6a2 _{– 8a c. 8p}3 _{– 24p}2 _{+ 16p}

4.

Critical Thinking

Use Algebraic Tiles to construct the model of 2x + 6. Then, construct a rectangle using the Algebraic Tiles and determine the factors of 2x + 6. Next, write down a procedure to determine the factors of any binomial.

5. Use Algebraic Tiles or draw a picture of rectangles representing the

following algebraic forms. Then write down the area of the rectangle as a multiplication of two binomials.

a. x2 _{+ 4x + 3 b. x}2 _{– 3x + 2 c. x}2 _{+ 3x – 4}

d. x2 _{+ 5x + 6 e. x}2 _{– 3x – 4 f. x}2 _{+ x – 2}

6.

Open Question

a. Draw two diagrams of Algebraic Tiles showing 6x2 _{+ 12x as a result of} multiplication. Place the unit x2 _{on the top left of the drawing.}

b. Write down the factors.

7. a. Factorise n2 _{– n}

8. Complete the following equations.

a. x2 _{– 6x – 7 = (x + 1)(x + …..) b. k}2 _{– 4k – 12 = (k – 6)(k + …..)}

c. t2 _{+ 7t + 10 = (t + 2)(t + …..) d. c}2 _{+ c – 2 = (c + 2)(c + …..)}

9.

Writing

Suppose you can factorise x2 _{+ bx + c as a multiplication of two}

binomials.

a. Explain what you know about the factors if c > 0 b. Explain what you know about the factors if c < 0

10. Factorise each of the following algebraic forms.

a. x2 _{+ 6x + 8 b. a}2 _{– 5a + 6 c. d}2 _{– 7d + 12}

d. t2 _{+ 7t – 18 e. x}2 _{+ 12x + 35 f. y}2 _{– 10y + 16}

11.

Open Question

For each of the following problems, determine three different numbers to complete each algebraic form so that it can be factorised as a multiplication of two binomials. Then show the factors.

a. x2 _{– 3x – F b. x}2 _{+ x + F c. x}2 _{+ Fx + 12}

12. Factorise each of the following algebraic forms having the pattern ax2 _{+ bx} + c with a ≠ 1.

a. 2x2 _{– 15x + 7 b. 5x}2 _{– 2x – 7 c. 2x}2 _{– x – 3}

d. 8x2 _{– 14x + 3 e. 2x}2 _{– 11x – 21 f. 3x}2 _{+ 13x – 10}

13. Factorise each of the following algebraic forms.

a. x2 _{+ 2x + 1 b. t}2 _{– 144 c. x}2 _{– 18x + 81}

d. 15t2 _{– 15 e. x}2 _{– 49 f. a}2 _{+ 12a + 36}

g. 4x2 _{– 4x + 1 h. 16n}2 _{– 56n + 49 i. 9x}2 _{+ 6x + 1}

j. 9x2 _{– 6x + 1 k. 2g}2 _{+ 24g + 72 l. 2x}3 _{– 18x}

14. a. Algebraic form (2x + 4)2 _{is equal to 4x}2 _{+ F + 16. What is the middle}

term?

b. Complete the following statement.

15.

Writing

Resume the procedure for factorising perfect quadratic trinomials. Give at least two examples.

16.

Mental Calculating

Determine the result of each multiplication below using the difference of two squares.

Example: (17)(23) = (20 – 3)(20 + 3)

Write d o wn the fa c to rs in the fo rm (a – b) (a + b)

Multip ly .

Subtract. = 400 – 9

= 391

a. (27)(33) b. (19)(21) c. (43)(37) d. (29)(31)

e. (16)(24) f. (51)(49) g. (18)(22) h. (98)(102)

17. a.

Open Question

Write a trinomial of a perfect square form.

b. Explain how you know that the above trinomial is a perfect square. c. Write down a trinomial which is not a perfect square.

18. Factorise each of the following algebraic forms.

a. 4

1_m2 _–

91 b. 41p2 – 2p + 4

c. 9

1_n2 _–

251 d. 251 k2 – 56k + 9

19. a.

Critical Thinking

The form (t – 3)2 _{– 16 is the difference of two} squares. If the form is expressed as a2 _{– b}2_{, determine a and b.}