KS3

Rewrite expressions by taking out common factors

Part ofMathsExpressions and formulae

Key points

Three steps. The first step is fifteen a squared plus six a plus twenty one a b. Fifteen, six and twenty one are highlighted orange. Three arrows point from the first expression to the second expression. The first arrow points from fifteen a squared to numbers one, three, five, fifteen. The second arrow points from six a to numbers one, two, three, six. The third arrow points from twenty one a b to the numbers one, three, seven, twenty one. The three in each set of numbers are highlighted orange. The third step is HCF of fifteen, six and twenty one equals three. The three is shaded orange. — Image caption,
When the expression has no constants, the highest common factor (HCF) of the coefficients is found.

Essential knowledge for factorising includes understanding factors and (HCF), as well as having a good knowledge of .
An that has been rewritten by taking out common factors has been .
Factorising is the reverse process of expanding brackets.
A factorised expression is to the original expression. They are . The identity symbol (≡) is used to link the original expression and its equivalent factorised expression.
Understanding and using algebraic notation correctly is necessary to complete the factorising process properly.

Three steps. The first step is fifteen a squared plus six a plus twenty one a b. Fifteen, six and twenty one are highlighted orange. Three arrows point from the first expression to the second expression. The first arrow points from fifteen a squared to numbers one, three, five, fifteen. The second arrow points from six a to numbers one, two, three, six. The third arrow points from twenty one a b to the numbers one, three, seven, twenty one. The three in each set of numbers are highlighted orange. The third step is HCF of fifteen, six and twenty one equals three. The three is shaded orange. — Image caption,
When the expression has no constants, the highest common factor (HCF) of the coefficients is found.

Finding the highest common factors (HCF) in an expression

When the includes a the only highest common factor (HCF) will be a number:
- Find the HCF of the and constant, the greatest factor that is common to all the numbers in the expression.
- The HCF for the expression is a number.
When the expression has no constants:
- Find the HCF of the coefficients, the greatest factor that is common to all the coefficients in the expression.
- Find common factors in each term, which may be a single variable or a combination of variables.
- The HCF of the expression is the of the number and variable HCFs.

Examples

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Image caption,
Find the highest common factor (HCF) of this expression.
Image caption,
The expression includes a constant (12). The number 28 is a coefficient. The HCF of the expression will be a number. Find the HCF of the numbers 28 and 12
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To find the HCF of the expression, work out the factors of each number. The factors of 28 are 1, 2, 4, 7, 14 and 28. The factors of 12 are 1, 2, 3, 4, 6 and 12. The highest common factor of 28 and 12 is 4. The HCF of 28𝒂 + 12 is 4
Image caption,
Find the HCF of this expression.
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The expression has no constants. The HCF of the coefficients is the greatest factor that is common to all the numbers in the expression. No written coefficient means the coefficient is 1. The coefficients are 3, 5 and 1 (𝒃 is 1𝒃). The factors of 3 are 1 and 3. The factors of 5 are 1 and 5. The factors of 1 (𝒃 is 1𝒃) is 1. The HCF of 3𝒂 + 5𝒂𝒃 + 𝒃 is 1
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There are no common variable factors. 3𝒂, 5𝒂𝒃 and 𝒃 do not have a variable that is common to all of them. The variable 𝒂 is in two of the terms, not all three. The variable 𝒃 is in two of the terms, not all three. The HCF of this expression is 1
Image caption,
Find the HCF of this expression.
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The expression has no constants. The HCF of the coefficients is the greatest factor that is common to all the numbers in the expression. The coefficients are 15, 6 and 21. The factors of 15 are 1, 3, 5 and 15. The factors of 6 are 1, 2, 3 and 6. The factors of 21 are 1, 3, 7 and 21. The HCF of 15, 6 and 21 is 3
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The common variable in 15𝒂² + 6𝒂 + 21𝒂𝒃 is 𝒂. The variable HCF is 𝒂
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The HCF of the expression is the number HCF (3) multiplied by the variable HCF (𝒂). 3 × 𝒂 = 3𝒂. The HCF of the expression is 3𝒂

Question

Find the highest common factor of this expression.

Twenty four n squared p minus eighteen p squared

Factorise simple expressions

When the expression includes a constant the highest common factor (HCF) is a number.
When the expression does not include a constant the HCF is a number, a variable or a combination of a number and variables.

To factorise an expression:

Find the HCF of the numbers in the expression.
Find the HCF of the variables in the expression.
Find the product of the number and variable HCFs and write this term in front of the bracket.
Work out the terms in the bracket by completing the factor pair for each of the original terms in the expression.

The identity symbol (≡) is used to link the original expression and its equivalent factorised expression.

Examples

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Image caption,
Factorise 18𝒙 + 12
Image caption,
The expression includes a constant (12). The HCF is a number. The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 12 are 1, 2, 3, 4, 6 and 12. The highest common factor of 18 and 12 is 6. The HCF 6 is written in front of the bracket.
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The first term in the bracket is 3𝒙 because the factor pair for 18𝒙 is 6 × 3𝒙. The second term in the bracket is 2 because the factor pair for 12 is 6 × 2. 18𝒙 +12 is factorised as the equivalent expression 6(3𝒙+2).
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Factorise 12𝒂𝒃 – 3𝒂
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The highest common factor of 12 and 3 is 3. The HCF of the variable terms 𝒂𝒃 and 𝒂 is 𝒂. The HCF of 12𝒂𝒃 and 3𝒂 is the product of 3 and 𝒂 (3𝒂).
Image caption,
3𝒂 is the HCF of 12𝒂𝒃 and 3𝒂. 3 is the highest common factor of 12 and 3. 𝒂 is the highest common factor of 𝒂𝒃 and 𝒂. 3𝒂 is written in front of the bracket. The first term in the bracket is 4𝒃 because 3𝒂 × 4𝒃 = 12𝒂𝒃. The second term in the bracket is -1 because 3𝒂 × -1 = -3. 12𝒂𝒃-3𝒂 is factorised as the equivalent expression 3𝒂(4𝒃-1).
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Factorise the expression.
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The highest common factor of 6 and 12 is 6. Remember that 𝒙² is the same as 𝒙 multiplied by 𝒙. The HCF of the variable terms 𝒙²𝒚 and 𝒙𝒚 is 𝒙𝒚. The HCF of 6𝒙²𝒚 and 12𝒙𝒚 is 6𝒙𝒚
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The HCF of 6𝒙²𝒚 and 12𝒙𝒚 is 6𝒙𝒚. 6𝒙𝒚 is written in front of the bracket. The first term in the bracket is 𝒙 because 6𝒙²𝒚 is 6𝒙𝒚 × 𝒙. The second term in the bracket is 2 because 12𝒙𝒚 is 6𝒙𝒚 × 2. 6𝒙²𝒚 + 12𝒙𝒚 factorises to the equivalent expression 6𝒙𝒚(𝒙 + 2).

Question

Factorise the expression.

Thirty m squared n plus forty five m n squared

Practise rewriting expressions

Quiz

Practise rewriting expressions by taking out common factors with this quiz. You may need a pen and paper to help you with your answers.

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