How to Factorise Using the Highest Common Factor (HCF)

Key takeaways

  • The highest common factor (HCF) is the largest factor shared by every term. The common factors of 18 and 24 are 1, 2, 3 and 6, so their HCF is 6.
  • For variables with the same base, the HCF is the lowest power. The HCF of x2, x5 and x3 is x2, because x2 is part of each of the higher powers.
  • To factorise using the HCF, write the HCF in front of a set of brackets, divide each term by it and put the answers inside. For example, 10x + 15 = 5(2x + 3).
  • If the first term is negative, take out a negative HCF and flip every sign inside the brackets. For example, −15x3 + 25x2 − 5x = −5x(3x2 − 5x + 1).
  • An expression is fully factorised when the terms left in the brackets are coprime, which means their only common factor is 1.

In this video

  1. 0:00 Introduction: factorising polynomials
  2. 1:05 Factors and factor pairs
  3. 6:15 Common factors and the HCF
  4. 15:27 Prime factorisation with factor trees
  5. 19:09 When the HCF is 1: coprime numbers
  6. 23:38 HCF of variables: the lowest power
  7. 27:20 HCF of algebraic terms
  8. 35:13 How to factorise using the HCF
  9. 37:01 Example 1: 10x + 15
  10. 39:35 Example 2: 36x − 28
  11. 44:58 Example 3: a negative first term
  12. 50:52 Example 4: fractions in every term

This lesson shows you how to find the highest common factor (HCF) of numbers and algebraic terms, then use it to factorise expressions. It is Year 10 content extended a little, and it builds the algebra you need for Mathematical Methods or Specialist Mathematics. The HCF is the first factorising method in the series, and the next lesson covers factorising by grouping.

What is a factor?

Watch this part from 1:05

Factors are numbers or variables that multiply together to make a term. They always multiply to make the term. They never add to make it.

Factors come in pairs. 21 = 1 × 21 and 21 = 3 × 7, so the factors of 21 are 1, 3, 7 and 21.

Factors of a variable

The index law am × an = am + n says that when you multiply terms with the same base, the powers add. So x9 = x2 × x7, and x9 = x4 × x5.

Any lower power of x is a factor of x9, and so is x9 itself. The factors of x9 are 1, x, x2, x3 and so on, up to x9. 1 is a factor of every number and every variable.

What is the highest common factor (HCF)?

Watch this part from 6:15

A common factor is a factor that two or more terms share. The highest common factor (HCF) is the largest one.

To find the HCF of 18 and 24, list the factors of each number in pairs:

  1. Write 1 and the number itself as far apart as possible. Every other factor fills in between them.
  2. Try 2, then 3, and so on. When one divides the number, write it down with its partner.
  3. Stop when you reach a number that is already in your list.

Factors of 18: 1, 2, 3, 6, 9, 18.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.

The common factors are 1, 2, 3 and 6. The largest is 6, so the HCF of 18 and 24 is 6.

How do you find the HCF of two numbers?

Watch this part from 14:28

There are two methods. Take 20 and 50 as the example.

Method 1: list the factors

Factors of 20: 1, 2, 4, 5, 10, 20.

Factors of 50: 1, 2, 5, 10, 25, 50.

The largest number in both lists is 10, so the HCF is 10.

Method 2: prime factorisation with a factor tree

A factor tree breaks a number down until every number at the bottom is prime. Divide by 5 if you can, otherwise try 3, then 2.

20 = 5 × 4 = 22 × 5

50 = 10 × 5 = 2 × 52

Take each prime that appears in both and choose its lowest power. That gives 2 and 5, so the HCF is 2 × 5 = 10.

Use listing for small numbers with only a few factors. Use prime factorisation for large numbers, where a full list gets long.

Three or more numbers

The process is the same. For 16, 28 and 40, a quicker way is to list only the factors of the smallest number, because nothing larger than 16 can go into 16. Then test them from the largest down. 16 and 8 don’t go into 28. 4 goes into 28 and into 40, so the HCF is 4.

What does coprime mean?

Watch this part from 19:09

The factors of 27 are 1, 3, 9 and 27. The factors of 28 are 1, 2, 4, 7, 14 and 28. The only factor they share is 1, so their HCF is 1.

Numbers whose only common factor is 1 are called coprime. When you factorise, you want the terms left inside the brackets to be coprime. Taking out a 1 changes nothing, so at that point you have factorised fully.

How do you find the HCF of variables?

Watch this part from 23:38

A common trap is to pick the highest power. For x2, x5 and x3, some students write x5. The HCF has to be common to every term.

x2 is x × x, and both x5 and x3 contain x × x. So the HCF is x2.

When the terms have the same base, the HCF is the lowest power. Think of three friends going out for lunch. One has $5, one has $10 and one has $15. The friends with $10 and $15 can still afford the $5 cafe. The friend with $5 can’t afford the $10 or $15 cafe. If they all want to eat together, they have to go to the $5 cafe. The lowest power works the same way, because it is the one that fits inside every term.

The HCF of y6, y4 and y7 is y4, because y4 × y2 = y6 and y4 × y3 = y7.

The HCF of a and a9 is a. Remember that a is really a1.

How do you find the HCF of algebraic terms?

Watch this part from 27:20

Treat the numbers and the variables separately, then multiply the pieces together.

Example: 12x4y2 and 18x2y5

  1. Numbers: the HCF of 12 and 18 is 6.
  2. The x terms: the lower power of x4 and x2 is x2.
  3. The y terms: the lower power of y2 and y5 is y2.
  4. The HCF is 6x2y2.

When a coefficient is 1

A term with no number in front has a coefficient of 1. For 14a2b3, ab3 and ab2, the numbers only share 1, and you don’t write it. The lowest powers are a and b2, so the HCF is ab2.

When you are finding the HCF, ignore any negative signs. You deal with them when you factorise.

How do you factorise an expression using the HCF?

Watch this part from 35:13

Factorising means dividing to create brackets. It is the opposite of expanding, where you multiply to remove brackets. You take the HCF out of every term and leave what remains inside the brackets.

  1. Find the HCF of all the terms.
  2. Write the HCF in front of a new set of brackets.
  3. Divide each term by the HCF and write each answer inside the brackets. Another way to think of it is to ask what you multiply the HCF by to get each term.
  4. Check by expanding it back, and make sure the terms in the brackets are coprime.

Step 1 is the hard part, and the sections above cover it. Write these steps in your notes and keep them with you while you practise.

How do you factorise 10x + 15?

Watch this part from 37:01

  1. The HCF of 10 and 15 is 5. The second term has no x, so the HCF is 5.
  2. Write 5 in front of a set of brackets.
  3. 10x ÷ 5 = 2x and 15 ÷ 5 = 3. Keep the plus sign.
  4. 2x and 3 are coprime, so you are done.

10x + 15 = 5(2x + 3)

Check by expanding:

5 × 2x + 5 × 3 = 10x + 15

You won’t check every time. Do it when you are unsure, or when marks are on the line in an exam or assessment.

How do you factorise 36x − 28?

Watch this part from 39:35

Use factor trees to find the HCF of 36 and 28.

36 = 3 × 12 = 32 × 22

28 = 7 × 4 = 22 × 7

The only prime they share is 2, and it appears as 22 in both. So the HCF is 22 = 4.

Work out what you multiply 4 by to get each term, and keep the minus sign:

4 × 9x = 36x

4 × 7 = 28

36x − 28 = 4(9x − 7)

Nothing except 1 divides both 9x and 7, so it is fully factorised.

How do you factorise when the first term is negative?

Watch this part from 44:58

Factorise −15x3 + 25x2 − 5x.

The HCF of 15, 25 and 5 is 5, and the lowest power of x is x. So the HCF is 5x.

5x(−3x2 + 5x − 1)

This is correct, but the brackets start with a negative term. You want the first term to be positive, so take out the negative HCF, −5x. Taking out a negative flips every sign inside the brackets:

−15x3 + 25x2 − 5x = −5x(3x2 − 5x + 1)

Both answers are equal. Take out the negative anyway, because it makes the later work easier.

Put the terms in order first

With one variable, write the polynomial from the highest power to the lowest, with the constant at the end, as in ax2 + bx + c. Then look at the first term only. If it is negative, take out a negative HCF. If it is positive, take out the HCF as normal.

With several variables, the order doesn’t matter. Put any constant at the end and check the sign of the first term.

How do you factorise when every term is a fraction?

Watch this part from 50:52

When every term has a fraction, take out a common fraction so no fractions are left in the brackets. Factorise (3/7)x6 + (1/7)x5 − (4/7)x4.

Every term has a denominator of 7. Strip each fraction back to a numerator of 1, and you can see 1/7 is a factor of 3/7, 1/7 and 4/7. The lowest power of x is x4. So take out (1/7)x4.

Multiplying a fraction by a whole number multiplies the numerator, so 1/7 × 3 = 3/7. Deal with the variables separately:

(1/7)x4 × 3x2 = (3/7)x6

(1/7)x4 × x = (1/7)x5

(1/7)x4 × (−4) = −(4/7)x4

(3/7)x6 + (1/7)x5 − (4/7)x4 = (1/7)x4(3x2 + x − 4)

Why can’t you multiply by 7 to clear the fractions?

That works in an equation, because you multiply both sides by 7, as you do when you solve equations with fractions. This is an expression with no equals sign, so multiplying it by 7 would change its value.

Frequently asked questions

What is the difference between factorising and expanding?

They are opposites. Expanding multiplies to remove brackets. Factorising divides a common factor out of every term and leaves what remains inside brackets.

Is the HCF of variables the highest power or the lowest power?

It is the lowest power, as long as the terms have the same base. The HCF of x2, x5 and x3 is x2.

How do you know when you have factorised fully?

Look at the terms left inside the brackets. If their only common factor is 1, they are coprime and the expression is fully factorised.

Still stuck on factorising?

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