Lesson 1.2: Expanding and Factorising

Introduction

In this lesson, we will explore the processes of expanding and factorising algebraic expressions. Understanding these concepts is crucial as you progress in your study of mathematics, as they form the foundation for more complex topics.

Learning Objectives

1. Expand single and double brackets and collect like terms.
2. Factorise quadratics, the difference of two squares, and common factors.
3. Recognise standard algebraic patterns.
4. Expand and simplify products of two or more brackets.
5. Factorise a range of quadratic and simple expressions.

Expanding Single Brackets

Expanding a single bracket involves multiplying a term outside the bracket by every term inside the bracket. The basic idea can be understood through the distributive property.

The Distributive Property

The distributive property states that for any numbers $a$, $b$, and $c$:

$$\text{If } a(b + c) = ab + ac$$

This means you multiply $a$ by both $b$ and $c$.

Example 1: Expanding a Single Bracket

Let’s expand the expression $3(x + 4)$:

1. Identify the term outside the bracket: $3$.
2. Multiply $3$ by $x$: $3 \times x = 3x$.
3. Multiply $3$ by $4$: $3 \times 4 = 12$.
4. Combine the results:

$$3(x + 4) = 3x + 12$$

Common Misconception

A common error is to either forget to multiply both terms or to add instead of multiply. Always remember to distribute the outer term to all inner terms.

Expanding Double Brackets

Expanding double brackets can be accomplished using the distributive property twice. We will learn to use the FOIL method, which stands for First, Outside, Inside, Last.

Example 2: Expanding Double Brackets

Let’s expand $(x + 2)(x + 3)$:

1. First: Multiply the first terms: $x \cdot x = x^2$.
2. Outside: Multiply the outside terms: $x \cdot 3 = 3x$.
3. Inside: Multiply the inside terms: $2 \cdot x = 2x$.
4. Last: Multiply the last terms: $2 \cdot 3 = 6$.
5. Combine the results:

$$ (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 $$

Collecting Like Terms

When expanding, you may end up with terms that can be combined, which is known as collecting like terms. For instance:

• In the example above, $3x$ and $2x$ are like terms and combine to make $5x$.

Factorising Expressions

Factorising is the reverse process of expanding. It means expressing an algebraic expression as a product of its factors. The common techniques include finding common factors, factorising quadratics, and identifying patterns such as the difference of squares.

Factorising Common Factors

To factor out a common factor, you find the highest common factor (HCF) in all terms and divide each term by this factor.

Example 3: Factorising Common Factors

Consider the expression $6x^2 + 9x$:

1. Identify the HCF: The HCF of $6x^2$ and $9x$ is $3x$.
2. Divide each term by $3x$:

• $6x^2 \div 3x = 2x$
• $9x \div 3x = 3$

1. Write the expression in factored form:

$$6x^2 + 9x = 3x(2x + 3)$$

Factorising Quadratics

Quadratic expressions are of the form $ax^2 + bx + c$. Factorising involves finding two binomials that multiply to give the original quadratic expression.

Example 4: Factorising a Quadratic Expression

Let’s factorise $x^2 + 5x + 6$:

1. Identify $a$, $b$, and $c$: Here, $a = 1$, $b = 5$, $c = 6$.
2. Look for two numbers that multiply to $c$ and add to $b$: The numbers are $2$ and $3$.
3. Write the factors:

$$(x + 2)(x + 3)$$

1. Check by expanding:

$$(x + 2)(x + 3) = x^2 + 5x + 6$$

This confirms our factorisation.

The Difference of Two Squares

This is a special pattern that can be factorised using the formula:

$$a^2 - b^2 = (a + b)(a - b)$$

Example 5: Factorising Using the Difference of Two Squares

Factorise $x^2 - 9$:

1. Identify $a^2$ and $b^2$: Here, $a^2 = x^2$ and $b^2 = 9 \implies b = 3$.
2. Apply the formula:

$$x^2 - 9 = (x + 3)(x - 3)$$

Conclusion

In this lesson, we have covered the essential skills of expanding and factorising algebraic expressions. You learned how to expand single and double brackets, collect like terms, and factorise using various methods, such as finding common factors and using standard algebraic identities.

Study Notes

• Expanding single brackets means to multiply the term outside the bracket with every term inside the bracket.
• Use the FOIL method for expanding double brackets: First, Outside, Inside, Last.
• To factorise, express the equation as a product of its factors.
• Factor common terms by finding the highest common factor (HCF).
• For quadratics, find two numbers that multiply to $c$ and add to $b$.
• The difference of two squares can be factored using $a^2 - b^2 = (a + b)(a - b)$.