GCSE Maths

Expanding and Factorising

Edexcel

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Introduction Expansion and Factorisation in Detail Solved Examples

Introduction

Expanding:

Simplifying the expression or equation by eliminating the brackets present. Multiplying the number outside the brackets with numbers inside.

Factorising:

Writing the simplified equation in a brief form means using brackets.

Expansion and Factorisation in Detail

Expansion:

Example:

$$2(3x + 5) + 7x$$

Identify the parenthesis in expression.

Apply the distributive property, multiply the term outside the parenthesis by the term inside.

$$6x + 10 + 7x$$

Combine the terms with same variable if any.

$$13x + 10$$

Final Answer: $13x + 10$

Factorisation:

Example:

$$5x + 10 + 3y + 9$$

Find the common factors for grouped terms

First group ($5x + 10$):
$5x = 5 \times x$
$10 = 5 \times 2$
Common Factor $= 5$

Second group ($3y + 9$):
$3y = 3 \times y$
$9 = 3 \times 3$
Common Factor $= 3$

Algebraic expressions 5x, 10, 3y, and 9 broken down into factors showing common factors

Combine the terms having common factors.

$$5(x + 2) + 3(y + 3)$$

Final Answer: $5(x + 2) + 3(y + 3)$

Expand the following algebraic equations:

These are also used as general formulae.

(1) Expansion of $(a+b)^2$

$$(a+b)^2 = (a+b)(a+b)$$ $$(a \times a) + (a \times b) + (b \times a) + (b \times b)$$ $$a^2 + ab + ab + b^2$$

$$(a+b)^2 = a^2 + 2ab + b^2$$

(2) Expansion of $(a-b)^2$

$$(a-b)^2 = (a-b)(a-b)$$ $$(a \times a) + (a \times -b) + (-b \times a) + (-b \times -b)$$ $$a^2 – ab – ab + b^2$$

$$(a-b)^2 = a^2 – 2ab + b^2$$

(3) Difference of Squares

$$(a + b)(a – b) = a^2 – b^2$$

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Solved Examples

Solved Example

Problem: Represent the multiplication of three consecutive integers in algebraic form. Write the expression by expanding it further.

SOLUTION

Suppose $x$ is an integer. The next two consecutive integers are $(x+1)$ and $(x+2)$.

Initial Representation:

$$x(x+1)(x+2)$$

Expansion (Step-by-step):

First, expand the two binomials:

$$x(x^2 + 2x + x + 2)$$

Simplify inside the brackets:

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

Multiply the $x$ through:

Final Answer: $x^3 + 3x^2 + 2x$

Solved Example

Problem: What will be the total price of 2 sets of 5 toffees and 6 chocolates if the price individually is represented by variables?

SOLUTION

Suppose the price of one toffee is $x$ and that of one chocolate is $y$, then the total price will be:

$$2(5x + 6y)$$

Expansion of expression 2 times 5x plus 6y shown using sweets, toffees, and chocolates

Final Answer: $10x + 12y$

Solved Example

Problem: Suppose there are 5 baskets of Apples and 6 baskets of Bananas. If Bananas are 15 more than Apples, represent them in algebraic form and expand.

Baskets filled with apples and bananas to represent grouping terms in factorisation

SOLUTION

If there are $x$ number of apples in one basket, then bananas will be $(x + 15)$. We can represent the total number of fruit as follows:

Initial Expression:

$$5x + 6(x + 15)$$

Expansion:

$$5x + 6x + 90$$

Final Answer: $11x + 90$

Solved Example

Problem: Suppose Person A owns $x$ cows, and 10 more horses than cows. Person B owns $y$ cows, and 15 more horses than cows. Represent the total number of animals they own.