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
- 3(2u – v)
- (3a+8b)5a
- -2n(7y – 4z)
Solutions
- 3(2u – v) = 3 x 2u – 3 x v
= 6u – 3v
- (3a + 8b)5a =3a x 5a + 8b x 5a
=15a2 + 40ab
- -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:
- 9a – 3z
- 5x2 + 15x
- Factorize
Solution:
- The HCF of 9a and 3z is 3
= 3(3a – z)
- The HCF of 5x2 and 15x is 5x
=
- Factorize
In the given expression,
Hence the products
have the factor in common. Thus,
CLASS ACTIVITY
- Remove brackets from the following:
- ii) iii)
- Factorize the following:
- 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:
- a) x2 + 5x + 2x + 10
- b) 2ab – 5a + 2b -5
c15 – xy + 5y – 3x
- 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
- 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.)
- 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
- Factorize the following quadratic expressions:
Solution
1st 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
- +10 and +1 +11
- +5 and +2 +7
- -10 and -1 -11
- -5 and -2 -7
Of these, only ii) gives the required result; Thus,
Note 1. The answer can be checked by expanding the brackets.
- The order of the brackets is not important.
1st 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
- +18 and +1 +19
- +9 and +2 +11
- +6 and +3 +9
Of these, only ii) gives the required result; Thus,
CLASS ACTIVITY
Factorize the following:
- 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:
- y2 – 4
- 36 – 9a2
- 5m2 – 45
Solution
- y2 – 4 = (y)2 – (2)2
= (y-2) (y+2)
- 36 – 9a2 = (6)2 – (3a)2
= (6-3a) (6+3a)
- 5m2 – 45 = 5(m2 – 9)
= 5(m2 – 32)
= 5(m+3)(m-3)
- =
=
CLASS ACTIVITY
Factorize the following:
- 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.
- a) 4x2 -5x + 7
- b) 1 + 4x – x2
Solution:
- a) i) +4 ii) -5
- b) i) -1 ii) +4
CLASS ACTIVITY
Write down the coefficient of i) x2 and ii) x in each of the following expansions:
- a) (x + 1) (x + 2) b) (2x – 1)2
SIMPLIFYING CALCULATIONS BY FACTORISATION AND SUBSTITUTION
EXAMPLE:
- 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
- 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
- Simplify the following by factorizing:
- a) 67 x 23 – 67 x 13
- b)
- 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.
- Area of washer =
=
=
But,
Area of washer =
= = 2
= 2 = 6.28
PRACTICE EXERCISE
- Expand and find the coefficient of d in the expansion of the following expressions?
- (d + 2) (d + 7) ii) (d – 8)(d + 3)
- Factorize the following:
- s2 + 10s +16 ii) n2 -7n + 10 iii) 36a2 -49b2 iv)
ASSIGNMENT
- Factorize the following:
- i) 10ax2 + 14 a2x ii) 3m – m(u – v) iii) h(2a – 7b) – 3k(2a – 7b)
- Factorize by grouping:
- i) ac – bd – bc + ad ii) 8a + 15by +12y +10ab
- Find the product of the coefficient of a2 and ab in the expansion of (a – b) (3a – 2b)
- Factorize: i) 25k2 – 16 ii) 5x2 – 45 c) y2 – 2y + 1
- 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
