Mathematics Form 1-4 : CHAPTER TEN – ALGEBRAIC EXPRESSION

Specific Objectives

By the end of the topic, the learner should be able to:

1. Use letters to represent numbers.
2. Write statements in algebraic form.
3. Simplify algebraic expressions.
4. Factorize algebraic expressions by grouping.
5. Remove brackets from algebraic expressions.
6. Evaluate algebraic expressions by substituting numerical values.
7. Apply algebra in real-life situations.

Content

1. Letters for numbers
2. Algebraic fractions
3. Simplification of algebraic expressions
4. Factorization by grouping
5. Removal of brackets
6. Substitution and evaluation
7. Problem solving in real-life situations

Introduction

An algebraic expression is a mathematical expression that consists of variables, numbers, and operations. The value of this expression can change depending on the values assigned to the variables. It is important to clarify these definitions and have students take notes on their graphic organizer to enhance understanding.

Note:

• Algebraic Expression — contains at least one variable, one number, and one operation. An example of an algebraic expression is n + 9.
• Variable — a letter used in place of a number. Sometimes, the variable is given a value which replaces it to solve the equation. Other times, the variable is unknown and the student solves the equation to find its value.
• Constant — a number that stands alone. For example, the 9 in n + 9 is a constant.
• Coefficient — a number placed in front of and attached to a variable. For example, in 5x + 3, the 5 is the coefficient.
• Term — each part of an expression separated by an operation. In n + 9, the terms are n and 9.

Examples

Write each phrase as an algebraic expression:

• Nine increased by a number r: 9 + r
• Fourteen decreased by a number x: 14 – x
• Six less than a number t: t – 6
• The product of 5 and a number n: 5 × n or 5n
• Thirty-two divided by a number y: 32 ÷ y

Example

An electrician charges sh 450 per hour and spends sh 200 a day on gasoline. Write an algebraic expression to represent his earnings for one day.

Solution: Let x represent the number of hours the electrician works in one day. The electrician’s earnings can be represented by the algebraic expression:

450x – 200

Simplification of Algebraic Expressions

Note:

Basic steps to follow when simplifying an algebraic expression:

• Remove parentheses by multiplying factors.
• Use exponent rules to remove parentheses in terms with exponents.
• Combine like terms by adding coefficients.
• Combine the constants.

Like and Unlike Terms

Like terms have the same variables raised to the same power, for example, 3b + 2b = 5b or a + 5a = 6a. These can be simplified further. Unlike terms have different variables, such as 3b + 2c or 4b + 2x, and cannot be simplified further.

Example

3a + 12b + 4a – 2b = 7a + 10b (collect the like terms)

2x – 5y + 3x – 7y + 3w = 5x – 12y + 3w

Example

Simplify: 2x – 6y – 4x + 5z – y

Solution:

2x – 6y – 4x + 5z – y = 2x – 4x – 6y – y + 5z

= (2x – 4x) – (6y + y) + 5z

= -2x – 7y + 5z

Note:

-6y – y = -(6y + y)

Example

Simplify:

Solution:

The L.C.M of 2, 3, and 4 is 12.

Therefore:

Example

Simplify:

Solution:

Example

5x² – 2x² = 3x²

4a²bc – 2a²bc = 2a²bc

a²b – 2c + 3a²b + c = 4a²b – c

Note:

Capital letters and small letters are not like terms.

Brackets

Brackets serve the same purpose as they do in arithmetic, indicating the order in which operations should be performed.

Example

Remove the brackets and simplify:

1. 3(a + b) – 2(a – b)
2. 1/3a + 3(5a + b – c)
3. 2b + 3(3 – 2(a – 5))

Solution

1. 3(a + b) – 2(a – b) = 3a + 3b – 2a + 2b = (3a – 2a) + (3b + 2b) = a + 5b
2. 1/3a + 3(5a + b – c) = 1/3a + 15a + 3b – 3c
3. 2b + 3{3 – 2(a – 5)} = 2b + 3{3 – 2a + 10} = 2b + 3(13 – 2a) = 2b + 39 – 6a = 2b + 39 – 6a

The process of removing the brackets is called expansion, while the reverse process of inserting the brackets is called factorization.

Example

Factorize the following:

1. 3m + 3n = 3(m + n) (the common term is 3, so we put it outside the bracket)

Solution

1. (3 is common)

Factorization by Grouping

When the terms of an expression that do not have a common factor are taken pairwise, a common factor can be found. This method is known as factorization by grouping.

Example

Factorize:

1. 3ab + 2b + 3ca + 2c
2. ab + bx – a – x

Solution

1. 3ab + 2b + 3ca + 2c = b(3a + 2) + c(3a + 2) = (3a + 2)(b + c)
2. ab + bx – a – x = b(a + x) – 1(a + x) = (a + x)(b – 1)

Algebraic Fractions

In algebra, fractions can be added and subtracted by finding the L.C.M of the denominators. This allows expressions to be combined into a single fraction.

Examples

Express each of the following as a single fraction:

Solution

= (10x – 10 + 5x + 10 + 4x)

=

Simplification by Factorization

Factorization is used to simplify expressions by breaking them down into products of simpler expressions.

Examples

Simplify p² – 2pq + q²

2p² – 3pq + q²

Solution

Numerator is solved first.

Then solve the denominator:

2p² – 2pq – pq – q² = (2p – q)(p – q)

(4m – 3n)(4m + 3n)

(4m + 3n)(m – n)

4m – 3n

m – n

Example

Simplify the expression.

Solution

Numerator: 18x(y – r)

Denominator: 9x(r – y)

Therefore:

Example

Simplify

Solution

=

Example

Simplify the expression completely.

Solution

=

Substitution

This is the process of giving variables specific values in an expression to evaluate it.

Example

Evaluate the expression if x = 2 and y = 1.

Solution

=

End of topic

Past KCSE Questions on the topic

1. Given that y = 2x – z, express x in terms of y and z.

x + 3z

1. Simplify the expression

x – 1 – 2x + 1

x 3x

Hence solve the equation

x – 1 – 2x + 1 = 2

x 3x 3

1. Factorize a² – b²

Hence find the exact value of 2557² – 2547²

1. Simplify p² – 2pq + q²

P³ – pq² + p²q – q³

1. Given that y = 2x – z, express x in terms of y and z.

Four farmers took their goats to a market. Mohammed had two more goats, Koech had 3 times as many goats as Mohammed, whereas Odupoy had 10 goats less than both Mohammed and Koech.

(i) Write a simplified algebraic expression with one variable, representing the total number of goats.

(ii) Three butchers bought all the goats and shared them equally. If each butcher got 17 goats, how many did Odupoy sell to the butchers?

1. Solve the equation

1 = 5 – 7

4x 6x

1. Simplify

a + b

2(a + b) 2(a – b)

1. Three years ago, Juma was three times as old as Ali. In two years’ time, the sum of their ages will be 62. Determine their ages.