FACTORIZATION: FACTORIZING EXPRESSION WITH A COMMON FACTOR BRACKET AND BY GROUPING

Subject:

MATHEMATICS

Term:

First Term

Week:

Week 4

Class:

JSS 3 / BASIC 9

Previous lesson: Pupils have previous knowledge of

COMPOUND INTEREST, PROFIT AND LOSS, AND SIMPLE INTEREST

that was taught in their previous lesson

Topic:

FACTORIZATION

Behavioural Objectives:

At the end of the lesson, learners will be able to

• Factorize algebraic expressions
• Factorize expressions with a common factor bracket and by grouping
• solve sums on special cases of factorization
• Us factorization to simplify expressions and coefficient of terms
• solve sums on word Problems Involving Factorizations

Instructional Materials:

• Wall charts
• Pictures
• Related Online Video
• Flash Cards

Methods of Teaching:

• Class Discussion
• Group Discussion
• Asking Questions
• Explanation
• Role Modelling
• Role Delegation

Reference Materials:

• Scheme of Work
• Online Information
• Textbooks
• Workbooks
• 9 Year Basic Education Curriculum
• Workbooks

CONTENT:

WEEK 4

TOPIC: FACTORIZATION

CONTENTS:

• Factorizing algebraic expressions
• Factorizing expressions with a common factor bracket and by grouping
• Special cases of factorization
• Using factorization to simplify expressions and coefficient of terms
• Word Problems Involving Factorization

REMOVING BRACKETS (REVISION)

Example 1

Remove brackets from

1. 3(2u – v)
2. (3a+8b)5a
3. -2n(7y – 4z)

Solutions

1. 3(2u – v) = 3 x 2u – 3 x v

= 6u – 3v

1. (3a + 8b)5a =3a x 5a + 8b x 5a

=15a² + 40ab

1. -2n(7y – 4z) =(-2n) x 7y – (-2n) x 4z

= -14ny + 8ny

FACTORIZATION BY TAKING COMMON FACTORS

To factorize an expression is to write it as a product of its factors

Example

Factorize the following:

1. 9a – 3z
2. 5x² + 15x
3. Factorize

Solution:

1. The HCF of 9a and 3z is 3

= 3(3a – z)

1. The HCF of 5x² and 15x is 5x

=

1. Factorize

In the given expression,

Hence the products

have the factor         in common. Thus,

CLASS ACTIVITY

1. Remove brackets from the following:
2. ii)   iii)
3. Factorize the following:
4. 5a +5z ii) 33bd-3de

FACTORIZATION BY GROUPING

There are a few different methods that can be used to factorize algebraic expressions. We will go over a few of the most common methods here with some examples.

One method is to use the distributive property. This states that for any numbers a, b, and c, we have:

a(b + c) = ab + ac

So, if we have an expression that is a product of two terms, one of which is a sum or difference, we can factorize it using the distributive property. For example:

2x(x + 5) = 2xx + 2×5

= 2x^2 + 10x

Another common method is to factorize by grouping. This can be used when we have an expression that is a sum or difference of two terms, both of which are products. For example:

3x^2 + 5x – 2 = (3x^2 + 5x) – 2

= (3x^2 + 5x – 2) + 2

= (3x^2 – 2) + (5x + 2)

= 3x(x – 1) + 5(x + 1)

yet another method is to use the factoring by zero property. This states that for any number a, we have:

a(b – c) = ab – ac

So, if we have an expression that is a product of two terms, one of which is a difference of two terms, we can factorize it using the factoring by zero property. For example:

2x(x – 5) = 2xx – 2×5

= 2x^2 – 10x

Example

Factorize the following:

Solution:

The terms and  have c in common.

The terms and  have  in common.

Grouping in pairs in this way,

=

The two products now have in common.

=

CLASS ACTIVITY

Factorize the following by grouping:

1. a) x² + 5x + 2x + 10
2. b) 2ab – 5a + 2b -5

c15 – xy + 5y – 3x

1. d) t + 6sz + 3s + 2tz

EXPANDING ALGEBRAIC EXPRESSIONS

Explain with five examples EXPANDING ALGEBRAIC EXPRESSIONS

1. What is an algebraic expression?

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators. Algebraic expressions are used to represent relationships between quantities in math equations or formulas.

2. What are the five examples of expanding algebraic expressions?

The five examples of expanding algebraic expressions are:

1) Distributive Property: a(b+c) = ab + ac

2) Combining Like Terms: a + b + a = 2a + b

3) FOIL Method: (a+b)(c+d) = ac + ad + bc + bd

4) Factoring: a(b+c) = ab + ac

5) Expanding Powers: (x+y)^2 = x^2 + 2xy + y^2

Example

1. Find the product of

(The expression means.

The product of is found by multiplying each term in the first bracket by each term in the second bracket.)

2. Expand (3a +2)².

Solution

CLASS ACTIVITY

1. What is the simplest form of the expression `(x^2+6x+9)/(x+3)`?

A. `x^2-6x+9`

B. `x-3`

C. `x^2/3+2x/3+3/3`

D. `x^2+2x+3`

2. What is the value of `(4x^2+12x-15)/(-3x-5)` when `x=-1`?

A. 2

B. -4

C. 4

D. -2

3. What is the value of `(x+2)(x-3)` when `x=4`?

A. 14

B. -5

C. 11

D. 7

4. What is the value of `(2x-1)/(x^2+3x-2)` when `x=-2`?

A. -1/5

B. 1/5

C. 5

D. -5

FACTORIZATION OF QUADRATIC EXPRESSIONS

A quadratic expression is one in which 2 is the highest power of the unknown(s) in the expression. For example,  are all quadratic expressions.

To factorize a quadratic expression is to express it as a product of its factors.

NOTE: A quadratic expression may not have other factors other than itself and 1. For example, this type of quadratic expression cannot be factorized.

 Example

1. Factorize the following quadratic expressions:

Solution

1^st step:

2nd step: Find two numbers such that their product is +10 and their sum is +7.

Number pair which has a product of +10                   sums of factors

1. +10 and +1 +11
2. +5 and +2 +7

• -10 and -1 -11

1. -5 and -2 -7

Of these, only ii) gives the required result; Thus,

Note          1. The answer can be checked by expanding the brackets.

2. The order of the brackets is not important.

1^st step:

2nd step: Find two numbers such that their product is +18 and their sum is +11. Since the 18 is positive, consider the 11 is positive, consider positive factors only

Factors of +10                                                 sums of factors

1. +18 and +1 +19
2. +9 and +2 +11

• +6 and +3 +9

Of these, only ii) gives the required result; Thus,

CLASS ACTIVITY

Factorize the following:

1. ii)

DIFFERENCE OF TWO SQUARES

The difference of the squares of two quantities is equal to the product of their sum and their difference.

1. What is the difference of two squares?

A. The difference of two squares is the square root of the sum of the squares of the two numbers.

B. The difference of two squares is the square root of the difference of the squares of the two numbers.

C. The difference of two squares is the product of the square of the first number and the second number.

D. The difference of two squares is the product of the square of the second number and the first number.

2. How do you solve a difference of two squares equation?

A. By factoring the equation into two linear factors.

B. By taking the square root of both sides of the equation.

C. By multiplying both sides of the equation by the square root of the first number.

D. By multiplying both sides of the equation by the square root of the second number.

3. What is the difference of two squares when the numbers are negative?

A. The difference of two squares is the same as when the numbers are positive.

B. The difference of two squares is the square root of the sum of the squares of the two numbers.

C. The difference of two squares is the square root of the difference of the squares of the two numbers.

D. There is no such thing as a negative difference of two squares.

Hence

Example:

Factorize the following:

1. y² – 4
2. 36 – 9a²
3. 5m² – 45

Solution

1. y² – 4 = (y)² – (2)²

= (y-2) (y+2)

1. 36 – 9a² = (6)² – (3a)²

= (6-3a) (6+3a)

1. 5m² – 45 = 5(m² – 9)

= 5(m² – 3²)

= 5(m+3)(m-3)

1. =

=

CLASS ACTIVITY

Factorize the following:

1. ii)             iii) 100 – w²

Coefficients of terms

The number before a letter in an algebraic expression or an equation is called its coefficient.

Example: Find the coefficient of i) x²; ii) x in each of the following expressions.

1. a) 4x² -5x + 7
2. b) 1 + 4x – x²

Solution:

1. a) i) +4 ii) -5
2. b) i) -1 ii) +4

CLASS ACTIVITY

Write down the coefficient of i) x² and ii) x in each of the following expansions:

1. a) (x + 1) (x + 2) b) (2x – 1)²

SIMPLIFYING CALCULATIONS BY FACTORISATION AND SUBSTITUTION

EXAMPLE:

1. Simplify 34 x 48 + 52 x 34

Solution:

34 is a common factor of 34 x 48 and 52 x 34.

Thus, 34 x 48 + 52 x 34 = 34(48 + 52)

= 34(100)

= 3400

2. Factorize the expression πr² + 2πrh. Hence, find the value of πr² + 2πrh when π = , r = 14 and h = 43.

Solution:

πr² + 2πrh = πr(r + 2h)

When π = , r = 14 and h = 43,

πr² + 2πrh =  x 14(14 + 2 x 43)

= 44(14 + 86)

= 44 x 100

= 4 400

CLASS ACTIVITY

1. Simplify the following by factorizing:
2. a) 67 x 23 – 67 x 13
3. b)
4. Factorize the expression 2πr² + 2πrh. Hence, find the value of the expression when π = , r=5, and h = 16.

WORD PROBLEMS INVOLVING FACTORIZATION

The figure below shows a circular metal washer. If the diameters of the washer and its hole are 3cm and 1cm respectively, find the area of the washer. Use the value 3.14 for .

SOLUTION

Let the outer and inner radii of the washer be R and r respectively.

1. Area of washer =

=

=

But,

Area of washer =

=  = 2

= 2  = 6.28

PRACTICE EXERCISE

1. Expand and find the coefficient of d in the expansion of the following expressions?
2. (d + 2) (d + 7) ii)         (d – 8)(d + 3)
3. Factorize the following:
4. s² + 10s +16 ii)         n² -7n + 10      iii)        36a² -49b²        iv)

ASSIGNMENT

1. Factorize the following:
2. i) 10ax² + 14 a²x ii) 3m – m(u – v) iii) h(2a – 7b) – 3k(2a – 7b)
3. Factorize by grouping:
4. i) ac – bd – bc + ad ii) 8a + 15by +12y +10ab
5. Find the product of the coefficient of a² and ab in the expansion of (a – b) (3a – 2b)
6. Factorize: i) 25k² – 16 ii) 5x² – 45 c) y² – 2y + 1
7. Use the difference of two squares to evaluate 96² – 4².

PRESENTATION:

Step 1:

The subject teacher revises the previous topic

Step 2:

He or she introduces the new topic

Step 3:

The class teacher allows the pupils to give their own examples and he corrects them when the needs arise

CONCLUSION:

The subject goes round to mark the pupil’s notes. He does the necessary corrections