An Algebraic Expression is a mathematical expression made up of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, division, powers, and brackets.
Unlike an equation, an algebraic expression does not contain an equals sign (=).
Algebraic expressions are used to represent unknown quantities, describe mathematical relationships, solve real-world problems, and simplify complex calculations.
They form the foundation of algebra and are widely used in mathematics, science, engineering, economics, computer programming, finance, and many other fields.
A Variable is a letter or symbol that represents an unknown or changing value in an algebraic expression.
Common variables include x, y, a, b, and n.
(i) In the expression 5x + 3, the variable is x.
(ii) In 8y − 12, the variable is y.
(iii) In 4a² + 7a − 9, the variable is a.
A Constant is a fixed numerical value that does not change.
Unlike variables, constants always have the same value.
(i) In 6x + 4, the constant is 4.
(ii) In 9y − 15, the constant is −15.
(iii) In 3a² + 5a + 8, the constant is 8.
A Coefficient is the numerical value that multiplies a variable.
If no number is written before a variable, its coefficient is understood to be 1.
(i) In 7x, the coefficient of x is 7.
(ii) In −4y, the coefficient of y is −4.
(iii) In x², the coefficient of x² is 1.
A Term is a single mathematical quantity in an algebraic expression.
Terms are separated by addition (+) or subtraction (−) signs.
Expression: 5x² + 3x − 7
The three terms are:
(i) 5x²
(ii) 3x
(iii) −7
Multiplication and division do not separate terms.
Like Terms are terms that have exactly the same variables raised to the same powers. Their coefficients may be different.
(i) 5x and 8x
(ii) 4y² and −9y²
(iii) 3ab and 7ab
Only like terms can be combined during simplification.
Unlike Terms are terms whose variables or exponents are different.
Unlike terms cannot be combined directly.
(i) 4x and 4y
(ii) 6x and 6x²
(iii) 5ab and 5a
A Monomial is an algebraic expression consisting of only one term.
(i) 8x
(ii) 5a²
(iii) −12
A Binomial is an algebraic expression consisting of exactly two terms.
(i) x + 5
(ii) 3y − 4
(iii) 2a² + 7
A Trinomial is an algebraic expression consisting of exactly three terms.
(i) x² + 4x + 4
(ii) 3a² − 2a + 9
(iii) 5m + 8n − 2
A Polynomial is an algebraic expression containing one or more terms with variables raised to whole-number exponents.
Monomials, binomials, and trinomials are all examples of polynomials.
(i) 4x³ − 2x² + 7x − 9
(ii) 6a⁴ + 3a² − a + 5
(iii) 9y⁵ − 2y³ + y − 1
Evaluating an Algebraic Expression means finding its numerical value by substituting the given value(s) of the variable(s) into the expression and then simplifying.
When evaluating an algebraic expression, always follow the correct order of operations (BODMAS/PEMDAS).
Evaluate the expression 3x + 5 when x = 4.
Step 1: Identify the given value.
x = 4
Step 2: Substitute the value into the expression.
3(4) + 5
Step 3: Simplify.
12 + 5 = 17
17
Final Answer: 17
Simplifying an Algebraic Expression means writing the expression in its simplest form by combining like terms and removing brackets where necessary.
Only like terms can be added or subtracted.
Simplify 5x + 3x − 2.
Step 1: Identify the like terms.
5x and 3x are like terms.
Step 2: Add the coefficients.
5x + 3x = 8x
Step 3: Write the simplified expression.
8x − 2
Final Answer: 8x − 2
Simplify 4a + 7 − 2a + 5.
Step 1: Group like terms.
(4a − 2a) + (7 + 5)
Step 2: Simplify each group.
2a + 12
Final Answer: 2a + 12
Simplify 3(x + 4).
Step 1: Apply the Distributive Law.
3 × x + 3 × 4
Step 2: Multiply.
3x + 12
Final Answer: 3x + 12
Simplify 5(2y − 3).
Step 1: Apply the Distributive Law.
5 × 2y − 5 × 3
Step 2: Multiply.
10y − 15
Final Answer: 10y − 15
Simplify 2(3x + 5) − x.
Step 1: Expand the bracket.
6x + 10 − x
Step 2: Combine like terms.
5x + 10
Final Answer: 5x + 10
Attempt the following Questions:
(i) Evaluate the algebraic expression 4x² − 3x + 2 when x = 3.
(ii) Simplify the algebraic expression 8a + 5 − 3a + 9 − 2.
(iii) A rectangle has a length of (3x + 5) metres and a width of (2x − 1) metres. Write an algebraic expression for the perimeter of the rectangle in its simplest form.
(iv) A stationery shop sells notebooks for $x each and pens for $2 each. If Mary buys 5 notebooks and 8 pens, write a simplified algebraic expression representing the total amount she pays.
(v) Expand and simplify the algebraic expression 4(2y − 3) + 5(y + 2).
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