Section B — Chapters 23–39

Algebraic Factorising: Introduction to Highest Common Factor (1st Year)

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to identify common factors in algebraic terms.
✓By the end of this lesson students will be able to find the Highest Common Factor (HCF) of algebraic terms.
✓By the end of this lesson students will be able to factorise simple algebraic expressions by taking out the HCF.
✓By the end of this lesson students will be able to understand factorising as the reverse process of multiplying out brackets.

Key concepts

Algebraic Factor

An algebraic factor is an algebraic term (a number, a letter, or a combination of both) that divides evenly into another algebraic term or expression. For example, '3' is a factor of '6x', and 'x' is a factor of 'x²'.

Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two or more algebraic terms is the largest algebraic term that divides evenly into all of them. To find the HCF, you find the HCF of the numerical coefficients and the lowest power of any common variables.

Factorising by HCF

Factorising is the process of writing an algebraic expression as a product of its factors, usually by taking out the Highest Common Factor (HCF). It is the reverse operation of multiplying out brackets. To factorise an expression like ax + ay, you identify the HCF (which is 'a' in this case) and write it outside a bracket, then divide each term in the original expression by the HCF to find the terms inside the bracket: a(x + y).

ax + ay = a(x + y)

Key facts to remember

1A factor is a number or algebraic term that divides another number or term exactly.
2The Highest Common Factor (HCF) is the largest factor common to all terms in an expression.
3Factorising is the process of writing an expression as a product of its factors, usually by taking out the HCF.
4Factorising is the reverse operation of multiplying out brackets.
5When finding the HCF, find the HCF of the numbers and the common letters separately.
6Always check your factorised answer by multiplying out the brackets to ensure it matches the original expression.

Worked examples

Example 1

Find the Highest Common Factor (HCF) of 12x and 18.

IIdentify the numerical coefficients: 12 and 18.

IIFind the HCF of 12 and 18. Factors of 12 are {1, 2, 3, 4, 6, 12}. Factors of 18 are {1, 2, 3, 6, 9, 18}. The HCF of 12 and 18 is 6.

IIIIdentify common variables: There is an 'x' in 12x but not in 18, so there are no common variables.

IVCombine the numerical HCF and common variables to get the overall HCF.

Answer

The HCF of 12x and 18 is 6.

Always look for the HCF of both the numbers and the letters separately.

Example 2

Factorise the expression 4x + 12.

IIdentify the terms: 4x and 12.

IIFind the HCF of the numerical coefficients (4 and 12): The HCF of 4 and 12 is 4.

IIIFind the HCF of the variables: 'x' is only in the first term, so there are no common variables.

IVThe overall HCF is 4.

VWrite the HCF outside a bracket: 4( ).

VIDivide each term in the original expression by the HCF: (4x ÷ 4) = x and (12 ÷ 4) = 3.

VIIPlace these results inside the bracket.

Answer

4(x + 3)

You can check your answer by multiplying out the bracket: 4(x + 3) = 4x + 12, which matches the original expression.

Example 3

Factorise the expression 5ab - 10a.

IIdentify the terms: 5ab and -10a.

IIFind the HCF of the numerical coefficients (5 and 10): The HCF of 5 and 10 is 5.

IIIFind the HCF of the variables: Both terms have 'a'. The lowest power of 'a' is a¹. There is 'b' in the first term but not the second, so 'b' is not common.

IVThe overall HCF is 5a.

VWrite the HCF outside a bracket: 5a( ).

VIDivide each term in the original expression by the HCF: (5ab ÷ 5a) = b and (-10a ÷ 5a) = -2.

VIIPlace these results inside the bracket.

Answer

5a(b - 2)

Pay close attention to the signs of the terms when dividing.

Common mistakes

✗Not finding the *highest* common factor (e.g., factorising 6x + 9 as 3(2x + 3) but then realising 3 is not the highest common factor if there was a larger one).
✗Forgetting to include a term inside the bracket, especially when a term is identical to the HCF (e.g., factorising 4x + 4 as 4(x) instead of 4(x + 1)).
✗Errors with signs when dividing terms by the HCF (e.g., 5a - 10 becomes 5(a + 2) instead of 5(a - 2)).
✗Only taking out a common numerical factor and forgetting common algebraic factors, or vice versa (e.g., factorising 3x² + 6x as 3(x² + 2x) instead of 3x(x + 2)).

Exam tips

★Always show your working step-by-step, especially when finding the HCF and dividing terms.
★Double-check your HCF before you write it outside the bracket to ensure it is indeed the *highest* common factor.
★After factorising, always multiply out your answer mentally or on paper to verify it matches the original expression.
★Practise identifying common factors quickly by looking at both the numerical coefficients and the variables in each term.

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