Algebraic expressions are mathematical phrases formed by combining numbers, variables and arithmetic operations. A variable is a letter that represents a quantity whose value can change, such as x, y or n. Constants are numbers whose value is fixed, such as 5, -3 or 1/2. An algebraic expression like 3x + 5 contains the variable term 3x and the constant term 5. Algebraic expressions allow us to write general rules, formulas and patterns in a compact way. For example, the perimeter of a square can be written as 4s, and the distance travelled at speed v for time t as v x t.
In Class 6 you learnt the idea of variables and simple expressions. In this chapter we study the parts of an expression, how to classify expressions by the number of terms, how to find the value of an expression, how to add and subtract expressions, and how to multiply a monomial by a monomial, a monomial by a binomial, and two binomials. These skills are the foundation of algebra, which is used everywhere in mathematics and science to express relationships and solve problems.
An expression is made of terms separated by plus or minus signs. In the expression 4x^2 + 3x - 7, there are three terms: 4x^2, 3x and -7. Each term is a product of factors. In the term 4x^2, the factors are 4 and x^2; in the term 3x, the factors are 3 and x. The numerical part of a term is called its coefficient. In 4x^2, the coefficient of x^2 is 4; in 3x, the coefficient of x is 3; the term -7 is a constant term with no variable.
Like terms have exactly the same variable parts, though their coefficients may differ. For example, 3x, 5x and -2x are like terms, and 7x^2 and -x^2 are like terms. Unlike terms have different variable parts, such as 4x and 4y, or 3x and 3x^2. Like terms can be combined by adding their coefficients; unlike terms cannot be added together.
An expression with one term is a monomial, such as 5x or -3y^2. An expression with two terms is a binomial, such as 3x + 4 or 2a - b. An expression with three terms is a trinomial, such as x^2 + 2x + 1. Expressions with more than three terms are simply called polynomials.
A polynomial in one variable is an expression with terms of non-negative whole number powers of a single variable. For example, x^2 + 3x + 2 is a polynomial in x, and the highest power of the variable is called the degree of the polynomial.
To find the value of an expression, substitute the given values of the variables and simplify using the rules of arithmetic. For example, the value of 3x + 5 when x = 2 is 3 x 2 + 5 = 6 + 5 = 11. The value of x^2 - 2x + 1 when x = 3 is 9 - 6 + 1 = 4. When substituting, always replace the variable with the value and follow the order of operations.
To add algebraic expressions, write the expressions in rows, align the like terms, and add the coefficients of the like terms. For example: (4x + 3) + (2x + 5) = (4x + 2x) + (3 + 5) = 6x + 8.
To subtract one expression from another, add the additive inverse: change the sign of each term of the expression being subtracted, then add. For example: (5x^2 + 3x - 2) - (2x^2 + x + 1) = 5x^2 + 3x - 2 - 2x^2 - x - 1 = 3x^2 + 2x - 3.
Only like terms can be combined. When combining 7x and -3x, add the coefficients: 7x - 3x = 4x. Terms with different variables, like 4x and 5y, remain as separate terms in the answer.
To multiply two monomials, multiply the coefficients and multiply the variable parts using the laws of exponents: (4x) x (3x) = (4 x 3) x (x x x) = 12x^2 (-2a) x (5b) = -10ab (3x^2) x (2x^3) = 6x^5
Use the distributive property. Multiply the monomial with each term of the other expression and add: 2x x (3x + 5) = 2x x 3x + 2x x 5 = 6x^2 + 10x x x (x^2 + 2x + 1) = x^3 + 2x^2 + x
To multiply two binomials, multiply each term of the first binomial by each term of the second binomial and then combine like terms: (x + 2)(x + 3) = x x x + x x 3 + 2 x x + 2 x 3 = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 (2a + 1)(a - 4) = 2a^2 - 8a + a - 4 = 2a^2 - 7a - 4
Any expression multiplied by zero gives zero. For example, 0 x (3x + 5) = 0.
| Type | Number of Terms | Example |
|---|---|---|
| Monomial | 1 term | 5x, -3y^2 |
| Binomial | 2 terms | 3x + 4, 2a - b |
| Trinomial | 3 terms | x^2 + 2x + 1 |
| Polynomial | Many terms | x^3 + x^2 + x + 1 |
| Type of Multiplication | Method | Example |
|---|---|---|
| Monomial x Monomial | Multiply coefficients and variables | (4x)(3x) = 12x^2 |
| Monomial x Binomial | Distributive property | 2x(3x + 5) = 6x^2 + 10x |
| Binomial x Binomial | Multiply every term of each | (x + 2)(x + 3) = x^2 + 5x + 6 |
| Any expression x 0 | Always zero | 0(3x + 5) = 0 |
Algebraic expressions bring variables, constants, coefficients and operations together into a general language for mathematics. Recognising terms, distinguishing like from unlike terms, evaluating expressions, and performing addition, subtraction and multiplication are the essential skills of this chapter. The distributive property unifies multiplication of monomials, binomials and trinomials, while the careful handling of signs prevents common errors. Algebraic expressions are the vocabulary of algebra: simple equations use them, and every higher topic, from factorisation to quadratic equations to coordinate geometry, builds directly on these foundations. Mastery here unlocks the rest of mathematics.