139. Factorising Expressions

Learning Intentions

• To understand that factorising is the reverse procedure of expanding
• find 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 $3(x+2)$
• 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 $3(x+4)=3x+12$ with reversing $3x+12$
b) compare $5(a-2)=5a-10$ with reversing $5a-10$

Worked Example 2

Find the highest common factor of each pair of terms:

a) $6x$ and $9x$
b) $8a$ and $12ab$
c) $15m$ and $20m$

Worked Example 3

Factorise each expression:

a) $6x+9$
b) $8a+12ab$
c) $10m-15$

Worked Example 4

Factorise each expression:

a) $12x+18x^2$
b) $14p-21pq$
c) $16ab+24a$

Worked Example 5

Factorise each expression fully:

a) $9x+6$
b) $20m-30n$
c) $25a^2+15a$

Worked Example 6

Write the expanded form, then factorise back:

a) $4(x+3)$
b) $5a-10$
c) $18x^2+12x$

Problems

Problem 1

State how factorising is the reverse of expanding:

a) compare $2(x+5)=2x+10$ with reversing $2x+10$
b) compare $4(y-3)=4y-12$ with reversing $4y-12$

Problem 2

Find the highest common factor of each pair of terms:

a) $4x$ and $10x$
b) $9b$ and $15bc$
c) $12m$ and $18m$

Problem 3

Factorise each expression:

a) $4x+10$
b) $9b+15bc$
c) $12m-18$

Problem 4

Factorise each expression:

a) $10x+15x^2$
b) $18p-24pq$
c) $21ab+14a$

Problem 5

Factorise each expression fully:

a) $8x+12$
b) $14m-21n$
c) $30a^2+20a$

Problem 6

Write the expanded form, then factorise back:

a) $3(x+4)$
b) $6y-18$
c) $24x^2+16x$

Exercises

Understanding and Fluency

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 ______

2. Find the highest common factor of each pair of terms:
   a) $6x$ and $12x$
   b) $8a$ and $20ab$
   c) $15m$ and $25m$
   d) $14p$ and $21pq$

3. Find the highest common factor of each pair of terms:
   a) $9x^2$ and $12x$
   b) $18ab$ and $24a$
   c) $16m$ and $40mn$
   d) $21pq$ and $14p$

4. Factorise each expression:
   a) $6x+12$
   b) $8a+16$
   c) $10m-15$
   d) $14p+21$

5. Factorise each expression:
   a) $12x+18x^2$
   b) $20a-30ab$
   c) $16m^2+24m$
   d) $15pq-10p$

6. Factorise each expression fully:
   a) $9x+3$
   b) $14a+28ab$
   c) $25m-20$
   d) $18p^2-12p$

7. Expand, then factorise back:
   a) $5(x+2)$
   b) $4(a-3)$
   c) $6m+12$
   d) $15x^2+10x$

8. Decide the highest common factor first, then factorise:
   a) $24x+36$
   b) $18a+27ab$
   c) $30m-45mn$
   d) $12p^2+20p$

Reasoning

9. Explain why factorising is called the reverse of expanding.

10. A student says that the highest common factor of $8x$ and $12x$ is $x$. Explain the mistake.

11. Noah says that $6x+12$ factorises to $6(x+12)$. Is he correct? Explain.

12. Explain why $15ab+10a$ can be factorised using $5a$ as a common factor.

13. A student factorises $12x+18$ as $3(12x+18)$. Describe the error.

Problem-solving

14. A rectangle has area $12x+20$. Factorise the expression to show a possible common width.

15. A builder uses $18m$ bricks on one wall and $24m$ bricks on another. Factorise the total number of bricks.

16. A pattern has $15x^2+10x$ tiles. Factorise this expression to show the common factor in each group.

17. A shop sells $14a$ apples and $21ab$ oranges in equal bundles. Factorise the expression for the total fruit.

18. A school arranges chairs in two groups of $20p$ and $30p^2$. Factorise the total number of chairs.

19. A gardener has two rectangular sections with areas $16x$ and $24$. 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 $6x+12$ factorises to $6x(1+2)$
• Students may forget that the bracket must expand back to the original expression
• Students may not check their factorised form by expanding it again