Expanding and Factorising Algebraic Expressions

TL;DR

Expanding means multiplying out brackets to remove them, making an expression longer. Factorising is the opposite: putting expressions into brackets by finding common factors. These are essential skills for simplifying and solving algebraic equations in higher math.

1. The Mental Model

Think of expanding and factorising as inverse operations, like adding and subtracting, or multiplying and dividing. Expanding "unpacks" an expression, while factorising "packs" it back up into a more compact form.

2. The Core Material

What is Expanding?

Expanding means getting rid of brackets by multiplying everything inside them by what's outside. It's like distributing a quantity across several items.

Expanding a single bracket:

You multiply the term outside the bracket by each term inside the bracket.

Example:
3(x + 5)

Here, you multiply 3 by x and 3 by 5.
3 * x = 3x
3 * 5 = 15
So, 3(x + 5) = 3x + 15

Expanding double brackets (Binomials):

This is often called FOIL: First, Outer, Inner, Last. It's a systematic way to make sure you multiply every term in the first bracket by every term in the second bracket.

Example:
(x + 2)(x + 3)

1. First: Multiply the first terms in each bracket: x * x = x²
2. Outer: Multiply the outer terms: x * 3 = 3x
3. Inner: Multiply the inner terms: 2 * x = 2x
4. Last: Multiply the last terms in each bracket: 2 * 3 = 6

Then, add all these results together and simplify by collecting like terms:
x² + 3x + 2x + 6 = x² + 5x + 6

What is Factorising?

Factorising is the reverse of expanding. You're looking to find common factors among the terms in an expression and 'pull them out' into a bracket.

Factorising by finding a common factor:

Look for the highest common factor (HCF) of all the terms. This HCF goes outside the bracket. Then, divide each original term by the HCF to get the terms inside the bracket.

Example:
4x + 12

1. Find the HCF of 4x and 12. The HCF of 4 and 12 is 4.
2. Write the HCF outside a bracket: 4(...)
3. Divide each original term by the HCF:
   4x / 4 = x
12 / 4 = 3
4. Put these results inside the bracket: 4(x + 3)

Factorising quadratic expressions (like x² + 5x + 6):

This usually means putting them into two brackets, like (x + a)(x + b). For a quadratic ax² + bx + c (where a=1), you need to find two numbers that:
1. Multiply to give c (the constant term).
2. Add to give b (the coefficient of x).

Example:
x² + 7x + 10

1. We need two numbers that multiply to 10 and add to 7.
2. Let's list factors of 10:
   1 * 10 (add to 11 – nope)
   2 * 5 (add to 7 – YES!)
3. So, the numbers are 2 and 5.
4. Put them into the brackets: (x + 2)(x + 5)

You can check your answer by expanding (x + 2)(x + 5) using FOIL.

Here's a diagram showing the relationship between expanding and factorising:

graph TD
    A["Algebraic Expression with Brackets"] --> B["Expand (Multiply Out)"]
    B --> C["Longer Expression (no brackets)"]
    C --> D["Factorise (Find Common Factors)"]
    D --> A
    B["Expand (Multiply Out)"] -- "Example: 3(x+5) = 3x+15" --> C
    D["Factorise (Find Common Factors)"] -- "Example: 3x+15 = 3(x+5)" --> A

3. Worked Example

Let's factorise 2x² + 10x - 12 and then check by expanding.

Step 1: Factorise by finding a common factor.
Notice that 2, 10, and 12 are all divisible by 2.
Pull 2 out as a common factor:
2(x² + 5x - 6)

Step 2: Factorise the quadratic expression inside the bracket.
We need two numbers that multiply to -6 and add to 5.
Let's consider factors of -6:
* 1 * -6 (adds to -5)
* -1 * 6 (adds to 5) - This is it!
* 2 * -3 (adds to -1)
* -2 * 3 (adds to 1)

So, the numbers are -1 and 6.
This means x² + 5x - 6 factorises to (x - 1)(x + 6).

Step 3: Combine the common factor with the new brackets.
2(x - 1)(x + 6)

Step 4: Check by expanding.
First, expand the two brackets: (x - 1)(x + 6)
* F: x * x = x²
* O: x * 6 = 6x
* I: -1 * x = -x
* L: -1 * 6 = -6
Combine: x² + 6x - x - 6 = x² + 5x - 6

Now, multiply this by the 2 outside:
2(x² + 5x - 6) = 2x² + 10x - 12

This matches our original expression, so our factorisation is correct!

4. Key Takeaways

• Expanding means multiplying terms to remove brackets, usually making the expression longer.
• Factorising means identifying common parts to put an expression into brackets, usually making it look more compact.
• FOIL helps you systematically expand two binomials (double brackets).
• Always look for the highest common factor first when factorising.
• When factorising x² + bx + c, find two numbers that multiply to c and add to b.
• Expanding and factorising are opposite operations, so you can use one to check the other.

Common mistakes to avoid:
* Forgetting to multiply all terms inside the bracket when expanding a single bracket.
* Only doing "First" and "Last" when expanding double brackets (forgetting Outer and Inner).
* Not finding the highest common factor, leaving a factor still inside the bracket.
* Getting the signs wrong when finding numbers that multiply to c and add to b.

5. Now Try It

For a 15-minute exercise, try these:
1. Expand: 4(2y - 3) + (y + 1)(y - 5)
2. Factorise fully: 3x² - 15x + 18

For success, you should get a single expanded expression with no brackets for the first one, and three factors (a number and two brackets) for the second, like A(x+B)(x+C). Remember to collect like terms and look for common factors at every step.