Algebraic Expressions

Algebraic Expressions

In this topic, you will learn about algebraic expressions, which are fundamental in mathematics. Understanding algebraic expressions is crucial because they form the basis for solving equations, working with polynomials, and performing calculus operations. This knowledge is essential for further studies in natural and engineering sciences.

1.1 Introduction to Algebra

Algebra is a branch of mathematics that uses symbols to represent numbers and relationships. These symbols allow us to formulate general statements or formulas that can be applied to various situations. For example, the formula for distance is given by:

D = s × t

where:

• D represents distance in kilometres.
• s represents speed in kilometres per hour.
• t represents time in hours.

This formula allows you to calculate any of the three variables if you know the other two. For instance, if you travel at 90 km/h for 3 hours, the distance travelled is:

D = 90 × 3 = 270 km.

1.2 Basic Components of Algebraic Expressions

An algebraic expression is a combination of variables, constants, and numbers, linked by various operations. Examples include:

• ax + b
• x² - 2x - 3
• (6x²y + 3y²)/2

In the expression 3x²y - 2xy + 7, the parts separated by + or - are called terms. The terms in this expression are:

• 3x²y
• -2xy
• 7

1.3 Terms, Coefficients, Variables, and Constants

In an algebraic expression, a term can be a number, a variable, or a combination of both. The coefficient is the number in front of a variable. For example, in the term 3x²y, 3 is the coefficient of x²y.

Variables are symbols that represent quantities that can change, while constants are fixed values. For instance, in the expression y = mx + c, m and c are constants, while x and y are variables.

1.4 Like and Unlike Terms

Terms can be classified as like or unlike:

• Like terms have the same variables raised to the same powers, e.g., 2x and 3x.
• Unlike terms have different variables or powers, e.g., 2x and 3y.

1.5 Substitution

Substitution involves replacing variables in an expression with specific values. For example, if you have the expression 3x² + 2x + 5 and substitute x = 2, you would calculate:

3(2)² + 2(2) + 5 = 3(4) + 4 + 5 = 12 + 4 + 5 = 21.

1.6 Working with Algebraic Expressions

1.6.1 Addition and Subtraction

To add or subtract algebraic expressions, you combine like terms. For example:

(2x + 3) + (3x + 4) = (2x + 3x) + (3 + 4) = 5x + 7.

1.6.2 Multiplication

To multiply algebraic expressions, apply the distributive property. For example:

(2x + 3)(x + 4) = 2x² + 8x + 3x + 12 = 2x² + 11x + 12.

1.6.3 Division

When dividing algebraic expressions, you may need to simplify first. For example:

(x² - 4)/(x - 2) = (x - 2)(x + 2)/(x - 2) = x + 2, for x ≠ 2.

1.7 Factorisation

Factorisation involves expressing an algebraic expression as a product of its factors. For example:

x² - 5x + 6 = (x - 2)(x - 3).

1.8 Special Product Formulas

Some products have specific forms that can be memorised for easier factorisation:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b²

1.9 Summary

In summary, algebraic expressions consist of variables, constants, and coefficients combined through operations. You can perform basic operations like addition, subtraction, multiplication, and division on these expressions. Factorisation and substitution are essential skills for manipulating algebraic expressions.