Expanding expressions with brackets
In the following algebraic expression where is the expansion and multiplication rules?
3y(4 – 2y) this can also be written as 3y × (4 – 2y)
Therefore the times × is not usually written in algebra. So when expanding or multiplying out the following expression, all the terms inside the bracket are multiplied by the term outside of it.
3y(4 – 2y) = 12y – 6y²
Expanding brackets and simplifying
There maybe scenarios where brackets will need to be expanded then simplified.
For example: 4x + 6x(5 – x)
This is expanded by:
4x + 6x(5 – x) = 4x + 30x – 6x²
Then it is simplified by
4x + 30x – 6x² = 34x – 6x²
Points to note:
- In the following example 4–(5n – 3), the minus (-) has the role of making all the terms inside the brackets as negative first.
- The same rules can be applied if there are more than one bracket in the equation
Expanding two brackets
When expanding two brackets, the rules in principle are the same but when tackling this scenario caution should be exercised.
For example, with the following algebraic expression:
(4 + t)(5 – 3t) = (4 + t) × (5 – 3t) but the × is not written in algebra.
To expand or multiply out this expression, every term in the first bracket should be multiplied by in the second bracket
(4 + t)(5 – 3t) = 4(5 – 3t) + t(5 – 3t)
=20 – 12t + 5t – 3t²
=20 – 17t – 3t² (this is also known as a quadratic expression)
Point to note:
- This example can be expanded and simplified in one step because in any expression in the form (x + a)(x + b), where a and b are fixed numbers, the expanded expression will have an x with a coefficient of a + b and the number at the end will be a × b.
The following example include squaring expressions:
(2 – 4a)² can also be written as: (2 – 4a) × (2 – 4a)
= (2 – 4a)(2 – 4a) = 2(2 – 4a) – 4a(2 – 4a)
= 4 – 8a – 8a – 16a²
= 4 – 16a – 16a²
(a + b)² = a² + 2ab + b²
The difference between two squares
The difference between two squares can be expressed as follows
(a + b)(a – b) = a² + b²