Welcome to Simplifying Expressions!

Have you ever looked at a messy room and felt much better once everything was tidied into the right boxes? Simplifying expressions in Algebra is exactly like that! We take a long, messy-looking string of letters and numbers and "tidy it up" so it’s much easier to read and work with.

In this chapter, you will learn how to group similar things together, how to open up brackets, and how to pack things back into brackets. These are the "golden rules" of Algebra that will make everything else you do in Maths much smoother.

1. The Basics: Vocabulary

Before we start tidying, we need to know what we are looking at. Algebra has its own names for things:

  • Expression: A group of numbers, letters, and operation signs (like \(+\) or \(-\)). Example: \(3x + 5y - 2\).
  • Term: A single part of an expression. In \(3x + 5y - 2\), the terms are \(3x\), \(5y\), and \(-2\).
  • Factor: Something that is multiplied. In the term \(3x\), the factors are \(3\) and \(x\).
  • Coefficient: The number in front of a letter. In \(7y\), the coefficient is \(7\).

Quick Review: Remember the "hidden" rules of notation:
\(ab\) means \(a \times b\)
\(3y\) means \(y + y + y\) or \(3 \times y\)
\(a^2\) means \(a \times a\)
\(\frac{a}{b}\) means \(a \div b\)

2. Collecting Like Terms

This is the most common way to simplify. To "collect like terms," you look for terms that have the exact same letter (or letters) and power. You can add or subtract them just like you would count apples and oranges.

Analogy: Imagine a fruit bowl with \(3\) apples, \(2\) bananas, and another \(4\) apples. You wouldn't say you have "\(3\) apples, \(2\) bananas, and \(4\) apples." You’d say you have "\(7\) apples and \(2\) bananas."

Example: Simplify \(5x + 3y + 2x - y\)

  1. Find the \(x\) terms: \(5x\) and \(+2x\). Together they make \(7x\).
  2. Find the \(y\) terms: \(3y\) and \(-y\). (Remember, \(-y\) is the same as \(-1y\)). Together they make \(2y\).
  3. Final answer: \(7x + 2y\)

Common Mistake to Avoid: You cannot combine different letters or powers. For example, \(x + x^2\) cannot be simplified further. They are like "apples" and "giant apples"—different things!

3. Multiplying a Single Term Over a Bracket

Sometimes an expression has a number or letter stuck to the outside of a bracket, like this: \(3(x + 4)\). This means "everything inside is being multiplied by the thing outside." This is also called expanding the bracket.

The Rule: Multiply the term on the outside by every single term on the inside.

Example: Expand \(5(2x - 3)\)

  • First part: \(5 \times 2x = 10x\)
  • Second part: \(5 \times -3 = -15\)
  • Result: \(10x - 15\)

Don't worry if this seems tricky at first! Just imagine the number outside the bracket is a delivery person who has to visit every "house" (term) inside the bracket.

4. Taking Out Common Factors (Factorising)

Factorising is the opposite of expanding. It’s like putting the "mess" back into a neat box. We look for the Highest Common Factor (the biggest number or letter that goes into all terms) and pull it outside the bracket.

Example: Factorise \(6x + 9\)

  1. What is the biggest number that goes into both \(6\) and \(9\)? It’s \(3\).
  2. Put the \(3\) outside the bracket: \(3( \quad )\)
  3. Divide the terms by \(3\) to see what's left: \(6x \div 3 = 2x\) and \(9 \div 3 = 3\).
  4. Final answer: \(3(2x + 3)\)

Key Takeaway: You can always check your factorising by expanding the brackets again. If you get the original expression, you’ve done it correctly!

5. Expanding Two Binomials

A "binomial" is just a fancy word for an expression with two terms, like \((x + 2)\). Sometimes you will be asked to multiply two of these together: \((x + 2)(x + 3)\).

A popular way to remember this is the FOIL method:

  • First: Multiply the first terms in each bracket.
  • Outside: Multiply the two outermost terms.
  • Inside: Multiply the two innermost terms.
  • Last: Multiply the last terms in each bracket.

Example: Expand \((x + 2)(x + 5)\)

  • F: \(x \times x = x^2\)
  • O: \(x \times 5 = 5x\)
  • I: \(2 \times x = 2x\)
  • L: \(2 \times 5 = 10\)

Now, collect the "like terms" (\(5x\) and \(2x\)):
Final answer: \(x^2 + 7x + 10\)

Did you know? This process is often called "Expanding Products." While you usually see two brackets, you might occasionally see three, which you solve by expanding the first two and then multiplying the result by the third!

Summary Checklist

Before you move on to Linear Equations or Formulae, make sure you are confident with these "Tidying Rules":

  • I can identify terms and factors.
  • I can add and subtract like terms.
  • I can expand a single bracket by multiplying everything inside.
  • I can factorise an expression by taking out the highest common factor.
  • I can expand two brackets using a method like FOIL or a grid.

Quick Tip: Always be careful with negative signs! Multiplying a negative number by another negative number makes a positive. That's the most common place where students lose marks.