Simplification of Algebraic Expressions

⭐ An expression formed using numbers, variables and mathematical operation signs is called an algebraic expression.

For example, 3x + 5, 2a² − 3a + 7, 5x + 3y − 2.

⭐ The process of using algebraic rules to write an expression in an equivalent but simpler form is called simplification of algebraic expressions.

📚 Many algebra problems become easier after simplifying the expression first. If you learn to identify like terms, remove brackets correctly, follow sign rules, and use identities, factorisation, laws of indices and BODMAS when needed, even complicated expressions can be simplified step by step.

👉 Important terms used in algebraic expressions

🔹 Term

Each part of an expression separated by a plus or minus sign is called a term. The + or − sign before a term belongs to that term.

For example, in 2a² − 3a + 7, the three terms are 2a², −3a and 7.

🔹 Variable

A letter whose value can change is called a variable.

For example, in 3x + 5, x is the variable.

🔹 Coefficient

The numerical factor multiplying a variable in a term is called its numerical coefficient.

For example, in 2x² − x + 7, the coefficient of 2x² is 2, and the coefficient of −x is −1.

🔹 Constant Term

A term containing no variable is called a constant term.

For example, in 2x² − x + 7, 7 is the constant term.

🔹 Like Terms

Terms having the same variables raised to the same powers are called like terms.

For example, 3x, 5x, −2x are like terms. Similarly, 2x²y and −7x²y are also like terms.

🔹 Unlike Terms

Terms with different variables or different powers of the variables are called unlike terms.

For example, 3x and 5y, or 2x and 3x², are unlike terms.

📚 Remember, 3x + 5x = 8x, but 3x + 5y cannot be simplified further. Similarly, 2x² + 3x contains unlike terms and cannot be combined into one term.

👉 Main methods of simplifying algebraic expressions

🔹 1. Combining like terms

The most common method of simplifying an algebraic expression is to combine like terms.

⭐ 5x + 3y + 2x − 4y

Write like terms together,

= 5x + 2x + 3y − 4y

= (5 + 2)x + (3 − 4)y

= 7x − y

⭐ 4a² + 3a − 2a² + 5a

= 4a² − 2a² + 3a + 5a

= 2a² + 8a

🔹 2. Removing brackets

When removing a bracket, the sign or factor before the bracket affects every term inside it.

⭐ 7x + (3x − 5)

= 7x + 3x − 5

= 10x − 5

⭐ 8a − (3a − 4)

There is a minus sign before the bracket, so the sign of each term inside changes.

= 8a − 3a + 4

= 5a + 4

👉 Common mistake

Writing 5x − (2x − 3) as 5x − 2x − 3 is incorrect.

The correct simplification is,

5x − (2x − 3) = 5x − 2x + 3 = 3x + 3.

🔹 3. Using the Distributive Law

a(b + c) = ab + ac and a(b − c) = ab − ac.

This means the factor outside the bracket must be multiplied by every term inside the bracket.

⭐ 3(x + 4)

= 3x + 12

⭐ 5(2x − 3)

= 10x − 15

⭐ 3(2x + 5) − 2(x − 4)

= 6x + 15 − 2x + 8

= 4x + 23

⭐ 4(2x − 3) − 3(x + 2)

= 8x − 12 − 3x − 6

= 5x − 18

🔹 4. Using Algebraic Identities

Some standard identities can simplify large expressions quickly.

Important identities:

🔸 (a + b)² = a² + 2ab + b²

🔸 (a − b)² = a² − 2ab + b²

🔸 (a + b)(a − b) = a² − b²

🔸 (x + a)(x + b) = x² + (a + b)x + ab

⭐ Example: Simplify (x + 3)² − (x − 3)².

Using identities,

= (x² + 6x + 9) − (x² − 6x + 9)

= x² + 6x + 9 − x² + 6x − 9

= 12x

📚 Another quick method is to use A² − B² = (A + B)(A − B).

🔹 5. Simplifying by factorisation

If the terms of an expression have a common factor, taking it outside the bracket gives a shorter form of the expression.

⭐ 8x² + 12x

The common factor of both terms is 4x.

Therefore, 8x² + 12x

= 4x(2x + 3)

⭐ 6a²b − 9ab²

The common factor is 3ab.

Therefore, 6a²b − 9ab²

= 3ab(2a − 3b)

🔹 6. Simplifying algebraic fractions

To simplify an algebraic fraction, factorise the numerator and denominator when needed. Then common factors in the numerator and denominator can be cancelled.

⭐

Factorising the numerator,

∵

= , where x ≠ −3.

= x − 3

⭐

=

= x + 2, where x ≠ 0.

👉 Important: In , the x in the numerator and the x in the denominator cannot be cancelled directly because x + 2 is a sum, not a common factor.

🔹 7. Using the laws of indices

For expressions containing powers, use the laws of indices.

For the same base,

🔸 a^m × aⁿ = a^m+n

🔸 , where a ≠ 0

🔸 (a^m)ⁿ = a^mn

⭐

= x⁴⁺²⁻²

= x⁴, where x ≠ 0.

📚 Remember, x² + x³ cannot be written as x⁵. The rule of adding indices applies only when powers with the same base are multiplied.

🔹 8. Simplifying Surds

A surd can often be simplified by separating a perfect-square factor.

⭐ √72

= √(36 × 2)

= 6√2

⭐ 3√8 + √18

= 3 × 2√2 + 3√2

= 6√2 + 3√2

= 9√2

📚 Only like surds can be combined. For example, 2√3 + 5√3 = 7√3, but √2 + √3 cannot be combined further.

🔹 Simplification using the BODMAS rule

When algebraic expressions contain numbers and several operations, the correct order of operations is important.

B O D M A S means:

🔸 O = Order or powers

📚 Remember: Division and multiplication have equal priority, so when both appear together, work from left to right. Similarly, addition and subtraction are also performed from left to right.

⭐ Simplify 2[3x − {x − (2x − 5)}].

Start with the innermost bracket,

Therefore,

= 2(3x + x − 5)

= 2(4x − 5)

= 8x − 10

⭐ A complete step-by-step example

Simplify 5(2x − 3) − 2(x + 4) + 3x.

First remove the brackets,

= 10x − 15 − 2x − 8 + 3x

Arrange like terms together,

= 10x − 2x + 3x − 15 − 8

= 11x − 23

👉 Quick check: If x = 2, both the original expression and 11x − 23 give −1. So the simplification is consistent.

⭐ Easy steps for simplification

When you see an algebraic expression, following this order will help reduce mistakes.

Step 1: Check whether there are brackets or powers.

Step 2: Start from the innermost bracket and remove brackets with the correct signs.

Step 3: Complete the required powers, multiplication and division.

Step 4: Identify the like terms.

Step 5: Arrange like terms together and add or subtract their coefficients.

Step 6: When needed, use identities, factorisation, algebraic fractions or laws of indices.

Step 7: Check whether the final answer can be simplified further.

Step 8: If possible, substitute a simple value and compare the original and simplified expressions.

👉 Avoid these common mistakes

🔸 3x + 2y = 5xy is incorrect. The two terms are unlike terms.

🔸 x² + x³ = x⁵ is incorrect. The rule for adding indices does not apply to addition.

🔸 −(a − b) = −a − b is incorrect; the correct result is −a + b.

🔸 a(b + c) = ab + c is incorrect; the correct result is ab + ac.

🔸 In fractions, cancel only common factors, not separate terms.

✍️ Try it yourself

🔹 1. Simplify 7x + 3x − 5.

🔹 2. Simplify 4a − (2a − 7).

🔹 3. Simplify 3(2x + 1) − 2(x − 4).

🔹 4.

🔹 5. Simplify .

🔹 6. Simplify .

🔹 7. Simplify √48.

🔹 8. Simplify 2[4x − {x − (x − 3)}].

Answers

🔸 1. 10x − 5

🔸 2. 2a + 7

🔸 3. 4x + 11

🔸 4. 16x

🔸 5. x − 4, where x ≠ −4

🔸 6. , where a ≠ 0

🔸 7. 4√3

🔸 8. 8x − 6

⭐ Important points to remember

🔸 Only like terms can be combined by addition or subtraction.

🔸 A negative sign belongs to its term. Therefore, in 2a² − 3a + 7, the second term is −3a.

🔸 When a minus sign comes before a bracket, the sign of every term inside the bracket changes.

🔸 In the distributive law, the outside factor must be multiplied by every term inside the bracket.

🔸 When powers with the same base are multiplied, their indices are added, but this rule does not apply to ordinary addition.

🔸 In an algebraic fraction, only common factors can be cancelled, and the denominator can never be zero.

🔸 Following BODMAS keeps the order of operations correct in complex expressions.

🔸 After finding the answer, check the signs and like terms once again.

📚 The main purpose of simplifying an algebraic expression is to write a complicated expression in an equivalent but easier form. The most important skills are identifying like terms, removing brackets correctly, following sign rules, and using the distributive law, identities, factorisation, laws of indices and BODMAS when needed. With regular step-by-step practice, even large and complex algebraic expressions become much easier to simplify.