Algebra Basics (Edexcel GCSE Maths): Revision Notes
Algebra basics
Getting started with algebra might seem daunting, but once you understand the fundamental rules and concepts, it becomes much more manageable. This section covers the essential building blocks you need to master before diving deeper into algebraic problems.
Understanding negative numbers
Negative numbers appear frequently in algebra, so it's crucial to understand how they behave in different operations. The key is remembering how signs interact when you multiply or divide.
Sign rules for multiplication and division
When working with negative numbers, these rules will help you determine whether your answer will be positive or negative:
Sign Rules for Multiplication and Division:
- Positive × Positive = Positive ()
- Positive × Negative = Negative ()
- Negative × Positive = Negative ()
- Negative × Negative = Positive ()
The same rules apply to division.
For example:
- (negative times positive gives negative)
- (negative times negative gives positive)
When two signs are together
Sometimes you'll encounter expressions where signs appear next to each other. Here's how to handle them:
- becomes (subtracting a negative is the same as adding)
- becomes (adding a negative is the same as subtracting)
Letters multiplied together
In algebra, we often need to multiply letters (variables) together. Understanding the shorthand notation makes expressions much cleaner and easier to work with.
Common algebraic notation
When letters are written next to each other, it means they're multiplied together:
Algebraic Multiplication Notation:
- means (the multiplication signs are left out for clarity)
- means (only the is squared, not the )
- means (the brackets mean both letters are squared)
- means (only the brackets are cubed)
Be careful with expressions like . This could mean either or . Usually, you'd interpret this as , but it's better to use brackets to be clear.
What are terms?
Before you can simplify algebraic expressions, you need to understand what makes up a term. This is fundamental to working with algebra effectively.
Definition of a Term: A term is a collection of numbers, letters, and brackets that are all multiplied or divided together. Terms are separated by plus and minus signs, and each term has either a positive or negative sign attached to it.
Looking at the expression , you can identify different types of terms:
- is an '' term
- is an '' term
- is a '' term
- is a '' term
- is a number term
If there's no sign in front of the first term, it means there's an invisible positive sign there.
Simplifying or collecting like terms
Simplifying algebraic expressions involves combining 'like terms' - terms that have the same combination of letters. This process makes expressions neater and easier to work with.
The process of simplifying expressions is systematic and follows a clear method that helps prevent mistakes and ensures you don't miss any terms.
Worked Example: Simplifying
Step 1: Put bubbles around each term Make sure you capture the positive or negative sign in front of each term. This helps you see what you're working with clearly.
Step 2: Rearrange terms in the best order Group like terms together so you can easily see which ones can be combined.
Step 3: Combine like terms Add or subtract the coefficients (numbers in front) of terms with the same letters.
- The terms:
- The number terms:
- Final answer:
This method works because you're essentially collecting all the terms together and all the number terms together, making the expression much simpler to work with.
Key Points to Remember:
- Negative × Negative = Positive, but Negative × Positive = Negative
- Letters written together (like ) mean they're multiplied together
- A term is everything multiplied/divided together, separated by + or - signs
- Like terms have the same combination of letters and can be combined
- Use the three-step method: bubble terms, rearrange, then combine like terms