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Factorising – Single Brackets
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Foundational skills
Find the HCF of two numbers
\[ \text{HCF of } 12 \text{ and } 18 \]
Find the largest number that divides exactly into two given numbers.
Find the HCF when one is a multiple
\[ \text{HCF of } 8 \text{ and } 24 \]
Find the HCF when one number divides exactly into the other.
Find the HCF of two algebraic terms
\[ \text{HCF of } 6x \text{ and } 9x^2 \]
Find the largest term that divides exactly into two algebraic terms.
Identify the common factor
\[ 4x + 8 \]
Look at an expression and identify what factor can be taken out.
Identify common factor with variable
\[ x^2 + 3x \]
Identify the common factor including any shared variables.
Two terms: numeric common factor
Numeric common factor (positive)
\[ 6x + 12 \]
Take out a common number where all terms are positive.
Numeric common factor (one negative)
\[ 8x – 12 \]
Take out a common number where one term is negative.
Numeric common factor (y variable)
\[ 9y + 15 \]
Take out a common number using a different variable.
Two terms: single variable
Factorise by taking out x
\[ x^2 + 5x \]
Take out x from an expression where every term contains x.
Taking out x (one negative)
\[ x^2 – 4x \]
Take out x when one term is negative.
Taking out x (negative first term)
\[ -x^2 + 3x \]
Take out x when the first term is negative.
Two terms: numeric and variable
Numeric and variable factor combined
\[ 4x^2 + 8x \]
Take out a factor that includes both a number and a variable.
Numeric and variable (one negative)
\[ 6x^2 – 9x \]
Take out a number and variable when one term is negative.
Larger coefficients
\[ 12x^2 + 18x \]
Take out a common factor from larger numbers.
Two terms: higher powers
Factorise by taking out x²
\[ x^3 + x^2 \]
Take out x² from an expression with x² or higher.
Taking out x² with coefficients
\[ 6x^3 + 9x^2 \]
Take out a term with x² and a numeric coefficient.
Factorise by taking out x³
\[ 2x^4 – 8x^3 \]
Take out x³ with its coefficient from higher powers.
Two terms: multiple variables
x as common factor (different variables)
\[ xy + xz \]
Take out x from terms with different second variables.
x with coefficients (different variables)
\[ 6xy + 9xz \]
Take out x with numeric coefficient from multi-variable terms.
xy as common factor
\[ x^2y + xy^2 \]
Take out xy from terms containing both x and y.
xy with coefficients
\[ 6x^2y + 9xy^2 \]
Take out xy with numeric coefficient from multi-variable terms.
Three or more terms
Three terms with numeric factor
\[ 6x + 12y + 18 \]
Take out a common number from three terms.
Three terms (mixed signs)
\[ 4x – 8y + 12 \]
Take out a common number when terms have mixed signs.
Three terms by taking out x
\[ x^3 + 2x^2 + 5x \]
Take out x from three terms that all contain x.
Three terms with numeric and x
\[ 4x^3 + 8x^2 + 12x \]
Take out both a number and x from three terms.
Negative leading coefficient
Factorise by taking out -1
\[ -x – 5 \]
Take out -1 from an expression where both terms are negative.
Taking out a negative number
\[ -6x – 12 \]
Take out a negative common factor from an expression.
Factorise by taking out -x
\[ -x^2 – 3x \]
Take out -x from an expression with negative terms.
Negative algebraic factor
\[ -6x^2 – 9x \]
Take out a negative factor including number and variable.
Simplify then factorise
Collect like terms then factorise
\[ 3x + 2x + 10 \]
Simplify by collecting like terms, then factorise.
Expand then factorise
\[ 2(x + 3) + 4x \]
Expand a bracket first, then factorise the result.
Expand two brackets then factorise
\[ 3(x + 2) + 2(x + 4) \]
Expand two brackets, collect like terms, then factorise.
Special cases
Factorise fully (two-stage)
\[ 12x^2 + 18x \]
Factorise completely – find the full HCF.
Fractional coefficients
\[ \frac{1}{2}x + \frac{3}{2} \]
Take out a common factor from fractions.
Four terms (common factor only)
\[ 2x + 4y + 6z + 8 \]
Take out a common factor from four terms.
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