Lesson 3.3: Factorising

Introduction

Welcome to Lesson 3.3 of Foundation Preparatory Mathematics, where we will explore the essential topic of factorisation. Factorising is a critical skill that allows us to simplify algebraic expressions and solve equations more efficiently. By the end of this lesson, you will be able to:

• Factorise by taking out a common factor.
• Factorise simple quadratics of the form $ x^2 + bx + c $.
• Recognise and factorise the difference of two squares.
• Factorise an expression by extracting the highest common factor.
• Factorise a simple quadratic into two brackets.

Let’s begin by understanding what factorisation means and how it can be applied.

What is Factorising?

Factorising is the process of breaking down an algebraic expression into factors that, when multiplied together, give the original expression. For example, if we have the expression $ x^2 + 5x + 6 $, we want to find two binomials that multiply to give this expression.

Concept of Common Factors

One way to factorise is by taking out common factors. A common factor is a number or term that divides all terms of an expression without leaving a remainder.

Example 1: Factorising by Taking Out a Common Factor

Consider the expression:

$$ 4x^2 + 8x $$

To factor this expression:

1. Identify the common factor in both terms $4x^2$ and $8x$. The common factor is $4x$.
2. Divide each term by $4x$:

• $ 4x^2 \div 4x = x $
• $ 8x \div 4x = 2 $

1. Rewrite the expression as:

$$ 4x^2 + 8x = 4x(x + 2) $$

Thus, the factorised form is $ 4x(x + 2) $.

Factorising Simple Quadratics

Quadratic expressions are of the form $ ax^2 + bx + c $. To factorise these, we look for two numbers that add to $ b $ and multiply to $ ac $ (the product of $ a $ and $ c $).

Example 2: Factorising a Simple Quadratic

Factor the expression:

$$ x^2 + 5x + 6 $$

1. Identify $ a = 1, b = 5, c = 6 $.
2. We need two numbers that add to $ 5 $ (the value of $ b $) and multiply to $ 6 $ ($ 1 \times 6 $). The numbers $ 2 $ and $ 3 $ satisfy this.
3. Write the quadratic in factored form:

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

Thus, the factorised form is $ (x + 2)(x + 3) $.

Recognising the Difference of Two Squares

The difference of two squares is a special case of factorisation where the expression is of the form $ a^2 - b^2 $. It can be factorised as:

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

Example 3: Factorising the Difference of Two Squares

Factor the expression:

$$ x^2 - 9 $$

1. Recognise that this is a difference of two squares:

• $ x^2 = (x)^2 $ and $ 9 = (3)^2 $.

1. Apply the formula:

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

Thus, the factorised form is $ (x + 3)(x - 3) $.

Extracting the Highest Common Factor

Sometimes an expression has a more complex structure that allows for extracting the highest common factor (HCF). The HCF is the largest factor that divides all terms in the expression.

Example 4: Factorising by Extracting the Highest Common Factor

Factor the expression:

$$ 6x^3 + 9x^2 $$

1. Identify the HCF. The HCF here is $ 3x^2 $.
2. Divide each term by $ 3x^2 $:

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

1. Write the factorised form:

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

Thus, the factorised form is $ 3x^2(2x + 3) $.

Factorising a Simple Quadratic into Two Brackets

As seen in examples above, quadratic expressions of the form $ ax^2 + bx + c $ can often be rearranged into the product of two linear factors. When $ a = 1 $, this process is straightforward; if a

eq 1 , it may require additional steps, such as factoring by grouping.

Example 5: Factorising a Non-Unit Quadratic

Factor the expression:

$$ 2x^2 + 7x + 3 $$

1. Identify $ a = 2, b = 7, c = 3 $. We search for two numbers that multiply to $ 2 \times 3 = 6 $ and add to $ 7 $. Those numbers are $ 6 $ and $ 1 $.
2. Rewrite the middle term using these numbers:

$$ 2x^2 + 6x + x + 3 $$

1. Group the terms:

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

1. Factor each group:

$$ 2x(x + 3) + 1(x + 3) $$

1. Factor out the common binomial:

$$ (2x + 1)(x + 3) $$

So, the factorised form is $ (2x + 1)(x + 3) $.

Conclusion

In this lesson, we have explored the process of factorising algebraic expressions, including taking out common factors, handling simple quadratics, and recognising special forms like the difference of squares. Factorisation simplifies the process of solving equations and understanding algebraic structures. Mastery of these skills will provide a strong foundation for future mathematical learning.

Study Notes

• Factorising means breaking down expressions into products of simpler factors.
• Identify common factors to simplify expressions.
• Factor simple quadratics using the product-sum relationship.
• Recognise patterns like the difference of squares for quick factorisation.
• Extract the highest common factor when applicable.
• Non-unit quadratics may require grouping techniques to factor.