139. Factorising Expressions

Learning Intentions

• To understand that factorising is the reverse procedure of expanding
• Solve the highest common factor of two terms
• factorise expressions

Pre-requisite Summary

• Know that expanding uses the distributive law to remove brackets
• Understand that multiplication can be written beside a bracket, for example
• Be able to Identify coefficients and pronumerals in terms
• Be able to find common factors of whole numbers
• Be able to identify like and unlike terms
• Understand that a factor is something multiplied to make a product

Worked Examples

Worked Example 1

State how factorising is the reverse of expanding:

a) compare with reversing

b) compare with reversing

Worked Example 2

Find the highest common factor of each pair of terms:

a) and

b) and

c) and

Worked Example 3

Factorise each expression:

a)

b)

c)

Worked Example 4

Factorise each expression:

a)

b)

c)

Worked Example 5

Factorise each expression fully:

a)

b)

c)

Worked Example 6

Write the expanded form, then factorise back:

a)

b)

c)

Problems

Problem 1

State how factorising is the reverse of expanding:

a) compare with reversing

b) compare with reversing

Problem 2

Find the highest common factor of each pair of terms:

a) and

b) and

c) and

Problem 3

Factorise each expression:

a)

b)

c)

Problem 4

Factorise each expression:

a)

b)

c)

Problem 5

Factorise each expression fully:

a)

b)

c)

Problem 6

Write the expanded form, then factorise back:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Complete each statement:

a) Factorising is the ______ of expanding

b) To factorise, look for a common ______

c) The highest common factor is the greatest factor shared by the ______

Exercise 2.

Find the highest common factor of each pair of terms:

a) and

b) and

c) and

d) and

Exercise 3.

Find the highest common factor of each pair of terms:

a) and

b) and

c) and

d) and

Exercise 4.

Factorise each expression:

a)

b)

c)

d)

Exercise 5.

Factorise each expression:

a)

b)

c)

d)

Exercise 6.

Factorise each expression fully:

a)

b)

c)

d)

Exercise 7.

Expand, then factorise back:

a)

b)

c)

d)

Exercise 8.

Decide the highest common factor first, then factorise:

a)

b)

c)

d)

Reasoning

Exercise 9.

Explain why factorising is called the reverse of expanding.

Exercise 10.

A student says that the highest common factor of and is . Explain the mistake.

Exercise 11.

Noah says that factorises to . Is he correct? Explain.

Exercise 12.

Explain why can be factorised Use as a common factor.

Exercise 13.

A student factorises as . Describe the error.

Problem-solving

Exercise 14.

A rectangle has area . Factorise the expression to show a possible common width.

Exercise 15.

A builder uses bricks on one wall and bricks on another. Factorise the total number of bricks.

Exercise 16.

A pattern has tiles. Factorise this expression to show the common factor in each group.

Exercise 17.

A shop sells apples and oranges in equal bundles. Factorise the expression for the total fruit.

Exercise 18.

A school arranges chairs in two groups of and . Factorise the total number of chairs.

Exercise 19.

A gardener has two rectangular sections with areas and . Factorise the total area to show a common dimension.

Potential Misunderstandings

• Students may think factorising and simplifying are always the same process
• Students may Choose a common factor that is not the highest common factor
• Students may ignore common pronumerals when finding the HCF
• Students may factor out only the number and forget the pronumeral part
• Students may place the wrong terms inside the bracket after factorising
• Students may think factorises to
• Students may forget that the bracket must expand back to the original expression
• Students may not Check their factorised form by expanding it again

Next: 140. Modelling Situations with Algebra