Multiplying Expressions (Junior Cert Mathematics): Revision Notes

Multiplying Algebraic Expressions

Multiplying algebraic expressions might sound tricky at first, but with some practice, it becomes much easier.
Think of it like multiplying numbers, except we also need to keep track of the variables (letters) that are part of the expression.
Let’s break it down step by step with examples similar to what you might see in your Junior Cycle Ordinary Level Maths exam.

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Key Concepts

Before looking at examples, let’s revise some basic ideas:

1. Coefficient – The number in front of a variable.
   Example: In 3x, the coefficient is 3.
2. Variable – A letter that represents an unknown number, such as x or y.
3. Exponent – The small number above and to the right of a variable showing how many times it’s multiplied by itself.
   Example: x² = x × x

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Rules for Multiplying Expressions

1. Multiply the coefficients – Multiply the numbers in front of the variables.
   (If there’s no number shown, it means the coefficient is 1.)
2. Multiply the variables – If the variables are the same, add their exponents.
   (If there’s no exponent, it means it’s 1.)
3. Combine the results – Write your final answer by joining the coefficient and variable together.

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Worked Example 1 – Multiplying Simple Terms

Problem: Multiply 3x × 4x

Step 1: Multiply the coefficients
3 × 4 = 12

Step 2: Multiply the variables
x × x = x²

Step 3: Combine
Final answer: 12x²

Explanation:
You multiply the numbers first, then add the powers of x (1 + 1 = 2).

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Worked Example 2 – Multiplying a Number by a Bracket (Distributive Law)

Problem: Expand 2(x + 5)

Step 1: Multiply the number outside by each term inside the bracket
2 × x = 2x
2 × 5 = 10

Step 2: Combine
Final answer: 2x + 10

Explanation:
This uses the distributive property, where every term inside the bracket is multiplied by the number outside.

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Worked Example 3 – Multiplying Two Brackets (Binomials)

Problem: Expand and simplify (x + 3)(x - 2)

Step 1: Multiply each term in the first bracket by each term in the second
x(x - 2) + 3(x - 2)

Step 2: Multiply each part
x × x = x²
x × -2 = -2x
3 × x = 3x
3 × -2 = -6

Step 3: Combine all terms
x² - 2x + 3x - 6

Step 4: Simplify like terms
x² + x - 6

Final Answer: x² + x - 6

Explanation:
Multiply carefully term by term, then combine any like terms to simplify your answer.

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Practice Problems

Try these yourself before checking your answers:

1. Multiply 5x × 3x
2. Expand 4(y + 7)
3. Expand and simplify (x + 4)(x - 3)
4. Expand 3(2x - 5)
5. Expand and simplify (2x + 1)(x - 6)

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Notes for Students

• Always multiply coefficients first before variables.
• Use the distributive law for brackets: multiply everything inside by what’s outside.
• When multiplying two brackets, multiply each term by each term and then combine like terms.
• Show all your steps — clear working helps earn method marks.