Multiplying brackets

Let’s first break down some jargon. In algebra:

  • Factors are the numbers, letters or expressions we’re multiplying together
  • Terms refer to individual parts of an expression, separated by plus or minus signs
  • Brackets () are the curved symbols we use to group factors or terms together

The Distributive Property

To multiply brackets, you need to know a rule known as the distributive property. It’s a fancy name for a simple concept. Imagine you have 5 \times (2 + 3).

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Here, you’d multiply 5 by both numbers inside the bracket: 5 \times 2 + 5 \times 3. The result is the same as if you added 2 + 3 first and then multiplied by 5.

Single Bracket Expansion

With single bracket expansions, you want to multiply into each term.

Basic example

Consider the example 3 \times (x + 2).

To expand this expression, you multiply the 3 by every term inside the bracket. That leads to 3 \times x + 3 \times 2, which simplifies to 3x + 6.

With Coefficients

Now, what if there’s a coefficient with the variable, like 2 \times (4x + 3)? The process is the same. Multiply 2 \times 4x + 2 \times 3 to get 8x + 6.

With Negative Terms

Be cautious when the bracket contains negative terms. For example, with 2 \times (x - 4), you will get 2 \times x + 2 \times (-4), which simplifies to 2x - 8. Notice how the negative sign got distributed as well.

Double bracket expansion (FOIL method)

Understanding FOIL

When dealing with two brackets, such as (x + 2) \times (x - 3), the FOIL method comes into play. FOIL stands for First, Outer, Inner, Last.

Step-by-Step Example

1. Multiply the First terms in each bracket, which gives you x \times x = x^2

2. Multiply the Outer terms: x \times -3 = -3x

3. Multiply the Inner terms: 2 \times x = 2x

4. Multiply the Last terms: 2 \times -3 = -6. Combine them to get x^2 - 3x + 2x - 6, which simplifies to x^2 - x - 6

Special Cases

Square of Binomials

A common special case is when a binomial (two terms separated by a plus or minus sign) is squared, like (x + y)^2. Don’t make the mistake of squaring just x and y individually.

Expand it as if it’s a double bracket, giving you x^2 + 2xy + y^2.

Coefficients and Multiple Terms

Sometimes you’ll encounter multiple terms or different coefficients inside a bracket. For example, 3 \times (x + y - 2z). You’ll need to distribute the 3 to each term inside the bracket, resulting in 3x + 3y - 6z.

Practice Questions

Question 1:

Expand: 4 \times (y + 1)

Solution

Distribute the 4 to each term inside the bracket, giving you 4 \times y + 4

Question 2:

Expand: 2 \times (3x - 5)

Solution

In this case, you distribute the 2 to each term inside the bracket. That would be 2 \times 3x + 2 \times (-5), which becomes 6x - 10 when simplified.

Question 3:

Expand: (x + 3) \times (x - 2)

Solution

Using the FOIL method, you first multiply the First terms (x \times x = x^2), then the Outer terms (x \times -2 = -2x), the Inner terms (3 \times x = 3x), and finally the Last terms (3 \times -2 = -6).

Combine all these together to get x^2 - 2x + 3x - 6, which simplifies to x^2 + x - 6.

Question 4:

Expand: (x - y)^2

Solution

This is a special case involving the square of a binomial. Expand it like a double bracket, treating it as (x - y) \times (x - y). Use the FOIL method to get x^2 - xy - xy + y^2. When simplified, this gives x^2 - 2xy + y^2.

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