Meaning of Algebra

The word Algebra comes from an Arabic word “Al-Jabr,” which means restoration or reunion of broken parts. In Mathematics, Algebra helps us to represent numbers and problems using letters and symbols. These letters can stand for numbers that are not yet known.

For example:

If we say, “A number added to 5 gives 12,”

we can represent the unknown number with a letter, say x.

So, x + 5 = 12.

This shows that Algebra helps us to write mathematical statements in short and simple forms using letters and numbers together.

Meaning of Algebraic Expression

An algebraic expression is a mathematical statement that contains numbers, letters (called variables), and signs of operation such as plus (+), minus (–), multiply (×), divide (÷), or powers (^).

It does not have an equal sign (=).

Examples of Algebraic Expressions

1. 3x + 2


2. 5y – 4


3. 2a + 3b – 7


4. 6m² + 5m + 2

Each of these examples is an algebraic expression because they have numbers and letters combined by mathematical signs, but they do not have an equal sign.

Let us take this expression as an example: 4x + 3y – 7

It has three main parts.

Terms

Each separate part joined by a plus (+) or minus (–) sign is called a term. So, in the expression 4x + 3y – 7, the terms are 4x, 3y, and –7.

Coefficient

The coefficient is the number that is multiplying a letter. In 4x, the coefficient is 4. In 3y, the coefficient is 3. If no number is written before the letter, it means the coefficient is 1. For example, in x + 2, the coefficient of x is 1.

Variables (or Unknowns)

The letters in an algebraic expression are called variables because their values can change. In 4x + 3y – 7, the variables are x and y.

Constant

A constant is a number that does not change. In 4x + 3y – 7, the –7 is the constant term.

Like Terms and Unlike Terms

When two or more terms have the same letters and the same powers, they are called like terms. When they do not have the same letters or powers, they are unlike terms.

Examples

1. 2x, 5x, and –7x are like terms because they all contain the same letter x.


2. 3ab and –8ab are like terms because they both have the letters a and b together.


3. 4x and 4y are unlike terms because their letters are different.


4. 3x² and 3x are unlike because x² is not the same as x.

Depending on the number of terms in the expression, we have different types.

1. Monomial: An expression that has only one term. Example: 3x, 5y², –7ab.


2. Binomial: An expression that has two terms. Example: x + 3, 2a – b, 4m + 7n.


3. Trinomial: An expression that has three terms. Example: a + b + c, 2x² + 3x + 4.


4. Polynomial: This is a general name for any algebraic expression that has two or more terms. Example: 2x³ + 5x² – x + 7.

To simplify means to make the expression shorter and neater by collecting like terms together.

Examples — Simplification of Algebraic Expressions

1. Problem: 4x + 7x − 3x
   
   
   1. Step 1: Identify like terms. All terms have the same variable x.
   
   
   2. Step 2: Add/subtract the coefficients: 4 + 7 − 3 = 8.
   
   
   3. Step 3: Write the result with the variable: 8x.
   
   


2. Problem: 2a + 3b + 5a − 2b
   
   
   1. Step 1: Group like terms: (2a + 5a) and (3b − 2b).
   
   
   2. Step 2: Add coefficients in each group: 2a + 5a = 7a; 3b − 2b = 1b.
   
   
   3. Step 3: Combine groups: 7a + b.
   
   


3. Problem: 6p + 4p − 3p
   
   
   1. Step 1: All terms are like terms (all contain p).
   
   
   2. Step 2: Add/subtract coefficients: 6 + 4 − 3 = 7.
   
   
   3. Step 3: Answer: 7p.
   
   


4. Problem: 9m + 2n − 5m + n
   
   
   1. Step 1: Group like terms: (9m − 5m) and (2n + n).
   
   
   2. Step 2: Compute each group: 9m − 5m = 4m; 2n + n = 3n.
   
   
   3. Step 3: Final simplified form: 4m + 3n.
   
   


5. Problem: 10x − 4x + 3y − 2y
   
   
   1. Step 1: Group like terms: (10x − 4x) and (3y − 2y).
   
   
   2. Step 2: Calculate: 10x − 4x = 6x; 3y − 2y = y.
   
   
   3. Step 3: Answer: 6x + y.
   
   


6. Problem: 7a + 5b − 2a + 3b
   
   
   1. Step 1: Group like terms: (7a − 2a) and (5b + 3b).
   
   
   2. Step 2: Add/subtract coefficients: 7a − 2a = 5a; 5b + 3b = 8b.
   
   
   3. Step 3: Final: 5a + 8b.
   
   

To expand means to remove brackets by multiplying the term outside the bracket by every term inside it.

Examples — Expansion (removing brackets)

1. Problem: 3(x + 4)
   
   
   1. Step 1: Multiply 3 by each term inside the bracket: 3×x and 3×4.
   
   
   2. Step 2: Compute: 3×x = 3x; 3×4 = 12.
   
   
   3. Step 3: Write expanded form: 3x + 12.
   
   


2. Problem: 2(a + b + c)
   
   
   1. Step 1: Multiply 2 by every term: 2×a, 2×b, 2×c.
   
   
   2. Step 2: Compute each product: 2a, 2b, 2c.
   
   
   3. Step 3: Result: 2a + 2b + 2c.
   
   


3. Problem: (a + 2)(b + 3)
   
   
   1. Step 1: Use each term of the first bracket to multiply every term of the second: a×b, a×3, 2×b, 2×3.
   
   
   2. Step 2: Compute: ab, 3a, 2b, 6.
   
   
   3. Step 3: Combine: ab + 3a + 2b + 6.
   
   


4. Problem: 5(p + 6)
   
   
   1. Step 1: Multiply 5 by p and 5 by 6.
   
   
   2. Step 2: 5p and 30.
   
   
   3. Step 3: Answer: 5p + 30.
   
   


5. Problem: 4(x + y + 2)
   
   
   1. Step 1: Multiply 4 by each term: 4x, 4y, 4×2.
   
   
   2. Step 2: Compute: 4x, 4y, 8.
   
   
   3. Step 3: Final: 4x + 4y + 8.
   
   


6. Problem: (m + 3)(n + 2)
   
   
   1. Step 1: Multiply m by n and 2, then 3 by n and 2: m×n, m×2, 3×n, 3×2.
   
   
   2. Step 2: Compute: mn, 2m, 3n, 6.
   
   
   3. Step 3: Combine: mn + 2m + 3n + 6.
   
   

To factorize means to write an expression as a product (multiplication) of its factors. It is the opposite of expansion.

Meaning of Factorization

The word factorization comes from the word factor, which means a number or term that divides another exactly without leaving a remainder.

In Mathematics, to factorize an algebraic expression means to write it as a product (multiplication) of two or more simpler expressions called factors.

Why We Factorize

We factorize algebraic expressions in order to:

1. Simplify long expressions into shorter forms.


2. Make it easier to solve equations.


3. Find common factors between terms.


4. Recognize hidden relationships in algebraic expressions.


5. Help in simplifying algebraic fractions and solving word problems.

Steps in Factorization

To factorize any algebraic expression, follow these simple steps:

1. Identify all the terms in the expression.


2. Find the common factor(s) (numerical and variable) in each term.


3. Divide each term by the common factor.


4. Write the common factor outside the bracket and the remaining parts inside the bracket.

It is the opposite of expansion.

When we expand, we multiply out the brackets.
When we factorize, we bring the common parts together into brackets.

Example:

If 2(x + 3) = 2x + 6
Then, factorizing 2x + 6 gives us back 2(x + 3).

So, expansion and factorization are opposite operations.

Difference Between Expansion and Factorization

Expansion	Factorization
It is the process of multiplying out brackets.	It is the process of putting brackets back.
It makes expressions longer.	It makes expressions shorter.
Example: 2(x + 3) = 2x + 6	Example: 2x + 6 = 2(x + 3)

Importance of Factorization

1. It helps to make long expressions simpler.


2. It helps in solving algebraic equations.


3. It makes it easy to simplify algebraic fractions.


4. It helps us to recognize patterns in algebra.


5. It prepares us for higher mathematics like quadratic equations.

Examples — Factorization (taking out common factor)

1. Problem: 6x + 9
   
   
   1. Step 1: Find highest common factor (HCF) of 6 and 9 → 3.
   
   
   2. Step 2: Divide each term by 3: 6x ÷ 3 = 2x; 9 ÷ 3 = 3.
   
   
   3. Step 3: Write factorized form: 3(2x + 3).
   
   


2. Problem: 4ab − 8ac
   
   
   1. Step 1: HCF for numbers is 4; both terms have a common letter a → HCF is 4a.
   
   
   2. Step 2: Divide: 4ab ÷ 4a = b; 8ac ÷ 4a = 2c.
   
   
   3. Step 3: Answer: 4a(b − 2c).
   
   


3. Problem: 9y + 6
   
   1. Step 1: HCF of 9 and 6 is 3.
   
   
   2. Step 2: Divide: 9y ÷ 3 = 3y; 6 ÷ 3 = 2.
   
   
   3. Step 3: Factorized: 3(3y + 2).
   
   


4. Problem: 12m² + 8m
   
   
   1. Step 1: HCF of numbers 12 and 8 is 4; both terms contain m so include m → HCF = 4m.
   
   
   2. Step 2: Divide: 12m² ÷ 4m = 3m; 8m ÷ 4m = 2.
   
   
   3. Step 3: Write: 4m(3m + 2).
   
   


5. Problem: 10p − 15q
   
   
   1. Step 1: HCF of 10 and 15 is 5. There are no common letters across both terms, so HCF = 5.
   
   
   2. Step 2: Divide: 10p ÷ 5 = 2p; 15q ÷ 5 = 3q.
   
   
   3. Step 3: Factorized form: 5(2p − 3q).
   
   


6. Problem: 14x + 21y
   
   
   1. Step 1: HCF of 14 and 21 is 7. No common variable in both terms, so HCF = 7.
   
   
   2. Step 2: Divide: 14x ÷ 7 = 2x; 21y ÷ 7 = 3y.
   
   
   3. Step 3: Answer: 7(2x + 3y).
   
   

Factorization by Grouping

Example 1

Factorize: ab + ac + xb + xc

1. Step 1: Group terms that have common factors: (ab + ac) + (xb + xc).


2. Step 2: Factorize each group separately: a(b + c) + x(b + c).


3. Step 3: Notice that (b + c) is common in both groups. Take (b + c) outside.


4. Answer: (a + x)(b + c).

Example 2

Factorize: 2xy + 4x + 3y + 6

1. Step 1: Group into two parts: (2xy + 4x) + (3y + 6).


2. Step 2: Factorize each group: 2x(y + 2) + 3(y + 2).


3. Step 3: (y + 2) is common. Take (y + 2) outside.


4. Answer: (2x + 3)(y + 2).

Substitution means putting given numbers in place of the letters (variables) in an expression.

Examples — Substitution (evaluating expressions)

1. Problem: If x = 2 and y = 3, evaluate 3x + 2y
   
   
   1. Step 1: Replace x with 2 and y with 3: 3(2) + 2(3).
   
   
   2. Step 2: Multiply: 3×2 = 6; 2×3 = 6.
   
   
   3. Step 3: Add: 6 + 6 = 12.
   
   


2. Problem: If a = 4 and b = 2, evaluate 2a + 3b
   
   
   1. Step 1: Substitute: 2(4) + 3(2).
   
   
   2. Step 2: Multiply: 8 + 6.
   
   
   3. Step 3: Add: 14.
   
   


3. Problem: If p = 5 and q = 1, evaluate p + 4q
   
   
   1. Step 1: Substitute: 5 + 4(1).
   
   
   2. Step 2: Multiply: 4×1 = 4.
   
   
   3. Step 3: Add: 5 + 4 = 9.
   
   


4. Problem: If m = 3 and n = 2, evaluate 2m + n
   
   
   1. Step 1: Substitute: 2(3) + 2.
   
   
   2. Step 2: Multiply: 6 + 2.
   
   
   3. Step 3: Sum: 8.
   
   


5. Problem: If r = 6 and s = 2, evaluate r − 3s
   
   
   1. Step 1: Substitute: 6 − 3(2).
   
   
   2. Step 2: Multiply: 3×2 = 6.
   
   
   3. Step 3: Subtract: 6 − 6 = 0.
   
   


6. Problem: If x = 1 and y = 4, evaluate 5x + y
   
   
   1. Step 1: Substitute: 5(1) + 4.
   
   
   2. Step 2: Multiply: 5 + 4.
   
   
   3. Step 3: Result: 9.
   
   

When an algebraic expression is written as a fraction, we call it an algebraic fraction. Example: 3x/4, (2a + 3)/5, 4x²/2y

We can add, subtract, multiply or divide algebraic fractions just like ordinary fractions, but we must make sure the denominators are not zero.

1. Problem: (3x/4) + (2x/4)
   
   
   1. Step 1: Denominators are same (4). Add numerators: 3x + 2x.
   
   
   2. Step 2: Add: 3x + 2x = 5x.
   
   
   3. Step 3: Final: (5x)/4.
   
   


2. Problem: (5a/6) − (2a/6)
   
   
   1. Step 1: Denominators equal → subtract numerators: 5a − 2a.
   
   
   2. Step 2: Compute: 5a − 2a = 3a.
   
   
   3. Step 3: Result: (3a)/6. Simplify by dividing numerator and denominator by 3 → a/2.
   
   


3. Problem: (x/3) × (6/2x)
   
   
   1. Step 1: Write the product: (x × 6) / (3 × 2x).
   
   
   2. Step 2: Simplify factors: numerator 6x; denominator 6x.
   
   
   3. Step 3: Cancel 6x/6x = 1 → final answer 1 (provided x ≠ 0).
   
   


4. Problem: (2m/5) + (3m/5)
   
   
   1. Step 1: Same denominator 5 → add numerators: 2m + 3m = 5m.
   
   
   2. Step 2: Result: (5m)/5 = m (since 5m ÷ 5 = m).
   
   
   3. Step 3: Final: m.
   
   


5. Problem: (4p/7) − (p/7)
   
   
   1. Step 1: Same denominator → subtract numerators: 4p − p = 3p.
   
   
   2. Step 2: Put over 7 → (3p)/7.
   
   
   3. Step 3: That is the simplified form (no further cancellation).
   
   


6. Problem: (3y/8) + (5y/8)
   
   
   1. Step 1: Same denominator → add numerators: 3y + 5y = 8y.
   
   
   2. Step 2: Expression: (8y)/8.
   
   
   3. Step 3: Simplify: (8y)/8 = y.
   
   

We can use algebra to represent real-life situations.

1. Problem: Musa is 5 years older than Tunde. If Tunde’s age is x, write Musa’s age.
   
   
   1. Step 1: The word “older than” means add 5 to Tunde’s age.
   
   
   2. Step 2: Tunde’s age = x, so Musa’s age = x + 5.
   
   
   3. Step 3: Final expression: x + 5.
   
   


2. Problem: Rectangle length = l, width = w. Write perimeter and area.
   
   
   1. Step 1: Perimeter is sum of all sides = l + w + l + w.
   
   
   2. Step 2: Combine like terms: 2l + 2w. Factor 2: 2(l + w).
   
   
   3. Step 3: Area = length × width = l × w → lw (or l × w).
   
   


3. Problem: A father is three times as old as his son. If the son’s age is y, the father’s age = ?
   
   
   1. Step 1: “Three times” means multiply by 3.
   
   
   2. Step 2: Son’s age = y, father’s age = 3 × y.
   
   
   3. Step 3: Final expression: 3y.
   
   


4. Problem: A book costs ₦x, a pen costs ₦y. Total cost of 3 books and 2 pens = ?
   
   
   1. Step 1: Cost of 3 books = 3 × x = 3x. Cost of 2 pens = 2 × y = 2y.
   
   
   2. Step 2: Add both costs: 3x + 2y.
   
   
   3. Step 3: Final algebraic expression: 3x + 2y.
   
   


5. Problem: A car travels at v km/h for t hours. Express the distance.
   
   
   1. Step 1: Distance = speed × time.
   
   
   2. Step 2: Speed = v, time = t, so distance = v × t.
   
   
   3. Step 3: Final: vt (or v×t).
   
   


6. Problem: Side of a square = s cm. Write expressions for perimeter and area.
   
   
   1. Step 1: Perimeter of square = 4 sides of length s → 4s.
   
   
   2. Step 2: Area = side × side = s × s = s².
   
   
   3. Step 3: Final expressions: Perimeter = 4s; Area = s².
   
   

Algebra is very useful in many areas of life and work. It helps us to:

1. Represent unknown quantities easily.


2. Find missing values in problems.


3. Make calculations shorter and easier to understand.


4. Apply mathematics to real-life situations, like business and science.


5. Form equations which we can solve to get exact answers.

1. Adding unlike terms together (for example, 3x + 4y = 7xy is wrong).


2. Forgetting to multiply all the terms inside the brackets when expanding.


3. Forgetting to take out the highest common factor when factorizing.


4. Putting wrong values during substitution.


5. Forgetting that there is no equal sign in an algebraic expression (only in equations).