198. Connecting Expansion and Factorisation

Learning Intentions

• Describe expansion and factorisation as reverse processes.
• Match monic quadratic expressions to equivalent binomial factors.
• Solve simple algebraic problems by choosing expansion or factorisation.

Pre-requisite Summary

• Know that expanding means multiplying out brackets.
• Know that factorising means writing an expression as a product of factors.
• Know that a monic quadratic has the form .
• Know that factor pairs for must multiply to and add to .
• Know how to expand binomial products such as .
• Know how to check factorisation by expanding the binomial factors.

Worked Examples

Worked Example 1

State whether each process is expansion or factorisation.

a)

b)

c)

Worked Example 2

Complete the reverse process.

a) Expand:

b) Factorise the result from part a).

c) Explain why the two expressions are equivalent.

Worked Example 3

Match each monic quadratic to its equivalent binomial factors.

a)

b)

c)

Factors:

Worked Example 4

Match each factorised form to its equivalent expanded form.

a)

b)

c)

Expanded forms:

Worked Example 5

Choose whether expansion or factorisation is more useful.

A rectangle has side lengths and .

a) Write an expression for the area.

b) Choose expansion or factorisation to write the area as a quadratic expression.

c) State the area expression.

Worked Example 6

Choose whether expansion or factorisation is more useful.

A rectangle has area .

a) Choose expansion or factorisation to find possible side lengths.

b) Write the side lengths as binomial factors.

c) Check by expanding.

Worked Example 7

Solve the algebraic problem by choosing expansion or factorisation.

A rectangular garden has area square metres.

a) Write possible expressions for the length and width.

b) If , find the numerical length and width.

c) Find the numerical area.

Problems

Problem 1

State whether each process is expansion or factorisation.

a)

b)

c)

Problem 2

Complete the reverse process.

a) Expand:

b) Factorise the result from part a).

c) Explain why the two expressions are equivalent.

Problem 3

Match each monic quadratic to its equivalent binomial factors.

a)

b)

c)

Factors:

Problem 4

Match each factorised form to its equivalent expanded form.

a)

b)

c)

Expanded forms:

Problem 5

Choose whether expansion or factorisation is more useful.

A rectangle has side lengths and .

a) Write an expression for the area.

b) Choose expansion or factorisation to write the area as a quadratic expression.

c) State the area expression.

Problem 6

Choose whether expansion or factorisation is more useful.

A rectangle has area .

a) Choose expansion or factorisation to find possible side lengths.

b) Write the side lengths as binomial factors.

c) Check by expanding.

Problem 7

Solve the algebraic problem by choosing expansion or factorisation.

A rectangular garden has area square metres.

a) Write possible expressions for the length and width.

b) If , find the numerical length and width.

c) Find the numerical area.

Exercises

Understanding and Fluency

Exercise 1

State whether each process is expansion or factorisation.

a)

b)

c)

d)

Exercise 2

Complete the reverse process.

a) Expand:

b) Factorise the expanded expression from part a).

c) Explain why the two forms are equivalent.

Exercise 3

Complete the reverse process.

a) Factorise:

b) Expand the factorised expression from part a).

c) Explain why the two forms are equivalent.

Exercise 4

Match each monic quadratic to its equivalent binomial factors.

a)

b)

c)

d)

Factors:

Exercise 5

Match each factorised form to its equivalent expanded form.

a)

b)

c)

d)

Expanded forms:

Exercise 6

Choose whether expansion or factorisation is more useful.

a) A rectangle has side lengths and . Find the area as a quadratic expression.

b) A rectangle has area . Find possible side lengths.

c) A square has side length . Find the area as a quadratic expression.

Exercise 7

Factorise each monic quadratic, then check by expanding.

a)

b)

c)

d)

Exercise 8

Expand each binomial product, then write the reverse factorisation statement.

a)

b)

c)

d)

Exercise 9

Copy and complete each equivalent statement.

a)

b)

c)

d)

Exercise 10

For each pair, decide whether the expressions are equivalent.

a) and

b) and

c) and

d) and

Reasoning

Exercise 11

Explain why expansion and factorisation are reverse processes.

Exercise 12

A student says:

“Factorising gives .”

Explain the mistake and write the correct factorised form.

Exercise 13

A student matches with .

a) Explain why the match is incorrect.

b) Write the correct binomial factors.

c) Check by expanding.

Exercise 14

Explain why is equivalent to .

Exercise 15

A student says that and are different because they look different.

Explain why the two forms are equivalent.

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) Expanding gives .

b) Factorising gives .

c) The expressions and are equivalent.

Problem-solving

Exercise 17

A rectangle has side lengths and .

a) Choose expansion or factorisation to find the area as a quadratic expression.

b) Find the area expression.

c) If , find the numerical area.

Exercise 18

A rectangle has area .

a) Choose expansion or factorisation to find possible side lengths.

b) Write the side lengths as binomial factors.

c) If , find the numerical length, width and area.

Exercise 19

A rectangular path has area .

a) Factorise the expression to find possible dimensions.

b) Check the dimensions by expanding.

c) Explain why factorisation was more useful than expansion.

Exercise 20

Create your own algebraic problem involving expansion and factorisation.

Your response must include:

• a quadratic expression or binomial product
• a decision about whether expansion or factorisation is more useful
• the completed algebraic working
• a sentence explaining how the two forms are equivalent

Potential Misunderstandings

• Students may think expansion and factorisation are unrelated procedures rather than reverse processes.
• Students may think factorisation means separating terms with addition signs rather than writing a product of factors.
• Students may not recognise that two expressions can be equivalent even when they look different.
• Students may match a quadratic to binomial factors by checking only the constant term.
• Students may find two numbers that multiply to but forget that they must also add to .
• Students may ignore negative signs when matching expressions such as .
• Students may choose expansion when factorisation is needed to find dimensions from an area expression.
• Students may choose factorisation when expansion is needed to find an area expression from side lengths.
• Students may forget to check factorisation by expanding the binomial factors and collecting like terms.