Expanding brackets Revision Notes for AQA GCSE Further Maths

Expanding brackets

What are quadratic expressions?

A quadratic expression is any mathematical expression that can be written in the form $ax^2 + bx + c$ , where the coefficient of $x^2$ is not zero. Understanding this concept is crucial because when we multiply two linear expressions together, we often get a quadratic result.

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The key feature that makes an expression quadratic is the presence of a squared term (like $x^2$ ). The coefficient of this squared term must not be zero, otherwise the expression would not truly be quadratic.

For example:

$x^2 + 3$ is a quadratic expression in $x$
$a^2$ is a quadratic expression in $a$
$2y^2 - 3y + 5$ is a quadratic expression in $y$

When you multiply two linear expressions like $(x + 5)(2x - 3)$ , the result becomes a quadratic expression. This happens because when you multiply the $x$ terms together, you get an $x^2$ term, which makes the entire expression quadratic.

Basic method for expanding brackets

The fundamental technique for expanding brackets involves using the distributive property. This means you multiply each term in the first bracket by each term in the second bracket, then combine any like terms.

Let's look at how this works with a straightforward example:

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Worked Example: Expanding $(x + 5)(2x - 3)$

When expanding this expression, you need to multiply every term in the first bracket by every term in the second bracket:

$(x + 5)(2x - 3) = x(2x - 3) + 5(2x - 3)$ $= 2x^2 - 3x + 10x - 15$
$= 2x^2 + 7x - 15$

This method involves multiplying everything in the second bracket by each term in the first bracket systematically. An alternative approach is to set up the multiplication in a grid format, which some students find easier to follow.

Expanding squared expressions

When you need to expand an expression that's been squared, like $(3x - 5)^2$ , it's important to remember that this means multiplying the expression by itself.

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A common mistake is to square each term individually instead of multiplying the entire expression by itself. Remember: $(3x - 5)^2 \neq (3x)^2 - 5^2$

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Worked Example: Expanding $(3x - 5)^2$

The key is to write the square as the product of two identical brackets, so you don't forget any terms:

$(3x - 5)^2 = (3x - 5)(3x - 5)$

Now you can apply the distributive method:

First, multiply the top line by $3x$ : $3x \times 3x = 9x^2$ and $3x \times (-5) = -15x$
Then, multiply the top line by $-5$ : $-5 \times 3x = -15x$ and $-5 \times (-5) = 25$
Finally, add all the products together: $9x^2 - 15x - 15x + 25 = 9x^2 - 30x + 25$

Multiplying polynomials of different degrees

Sometimes you'll need to multiply expressions that have more than two terms or involve higher powers of variables. The same distributive principle applies, but you need to be more systematic.

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Worked Example: Multiplying $(x^3 + 2x - 4)$ by $(x^2 - x + 3)$

For this type of multiplication, you multiply each term in the first polynomial by each term in the second polynomial:

$(x^3 + 2x - 4)(x^2 - x + 3)$

$= x^3(x^2 - x + 3) + 2x(x^2 - x + 3) - 4(x^2 - x + 3)$ $= x^5 - x^4 + 3x^3 + 2x^3 - 2x^2 + 6x - 4x^2 + 4x - 12$ $= x^5 - x^4 + 5x^3 - 6x^2 + 10x - 12$

The crucial step is combining like terms at the end to simplify your answer completely.

Expanding cubic expressions

When dealing with cubic expressions like $(a - 2)^3$ , it's often helpful to break the problem down into manageable steps rather than trying to expand everything at once.

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Worked Example: Expanding $(a - 2)^3$

Think of this as $(a - 2)^3 = (a - 2)(a - 2)^2$

Step 1: Work out $(a - 2)^2$ : $(a - 2)^2 = (a - 2)(a - 2) = a^2 - 2a - 2a + 4 = a^2 - 4a + 4$

Step 2: Then multiply this result by $(a - 2)$ : $(a - 2)^3 = (a - 2)(a^2 - 4a + 4)$ $= a(a^2 - 4a + 4) - 2(a^2 - 4a + 4)$ $= a^3 - 4a^2 + 4a - 2a^2 + 8a - 8$ $= a^3 - 6a^2 + 12a - 8$

This step-by-step approach helps prevent errors and makes the process more manageable.

Common exam tips and strategies

When expanding brackets in exams, these strategies will help you achieve accurate results:

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Essential Exam Strategies:

Take your time with the setup - Write out what you're multiplying clearly before you start
Use systematic multiplication - Work through each term methodically rather than jumping around
Check your algebra - Make sure you're handling negative signs correctly
Combine like terms carefully - This is where many errors occur, so double-check your arithmetic
Verify your degree - If you're multiplying two linear expressions, your answer should be quadratic

The most common mistakes happen when students rush through the multiplication or forget to combine like terms properly. Always take a moment to check that your final answer makes sense in terms of the degree of the polynomial you expect.

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Key Points to Remember:

Expanding brackets means multiplying every term in one bracket by every term in the other bracket
When two linear expressions are multiplied together, the result is always a quadratic expression
Squared expressions like $(3x - 5)^2$ should be written as $(3x - 5)(3x - 5)$ before expanding
For complex polynomials, work systematically through each multiplication and combine like terms at the end
Breaking down cubic expansions into steps (like squaring first, then multiplying) makes them much more manageable

Expanding brackets (AQA GCSE Further Maths): Revision Notes