mr barton maths

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Algebraic Fractions

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Foundational Skills
Identify the HCF of algebraic terms
\[ \text{HCF of } 6x^2, 9x \]
Find the highest common factor of two algebraic terms.
Recognise when a fraction can be simplified
\[ \frac{x+3}{x+5} \text{ simplify?} \]
Determine if an algebraic fraction can be simplified.
Write an integer as an algebraic fraction
\[ 5 \text{ with denom. } x+2 \]
Express an integer with a specified denominator.
Simplifying Algebraic Fractions (Monomials)
Simplify with numerical factors only
\[ \frac{6}{9x} \]
Simplify by cancelling common numerical factors.
Simplify with variable factors only
\[ \frac{x^3}{x^5} \]
Simplify using index laws for variables.
Simplify with single terms
\[ \frac{12x^2}{8x^5} \]
Simplify fractions with single algebraic terms.
Simplifying Algebraic Fractions (Linear)
Simplify already-factorised fractions
\[ \frac{3(x+2)}{6(x+2)} \]
Simplify fractions in factorised form.
Simplify by factorising numerator
\[ \frac{2x+6}{x+3} \]
Factorise the numerator first, then simplify.
Simplify by factorising denominator
\[ \frac{x-4}{3x-12} \]
Factorise the denominator first, then simplify.
Simplify by factorising both parts
\[ \frac{4x+8}{6x+12} \]
Factorise numerator and denominator, then simplify.
Simplifying Algebraic Fractions (Quadratic)
Simplify using difference of two squares
\[ \frac{x^2-9}{x+3} \]
Factorise using difference of squares pattern.
Simplify with quadratic factorisation
\[ \frac{x^2+5x+6}{x+2} \]
Factorise a monic quadratic, then simplify.
Simplify with two quadratics
\[ \frac{x^2-x-6}{x^2+x-2} \]
Factorise both quadratics, then simplify.
Simplify with sign reversal
\[ \frac{4-x}{x-4} \]
Recognise that (a-b) = -(b-a).
[Higher] Simplify non-monic quadratic
\[ \frac{2x^2+5x+3}{2x+3} \]
Factorise a non-monic quadratic first.
Multiplying Algebraic Fractions
Multiply with single terms
\[ \frac{2x}{3} \times \frac{9}{4x^2} \]
Multiply fractions with single terms and simplify.
Multiply with factorised expressions
\[ \frac{x+2}{3} \times \frac{6}{x+2} \]
Multiply fractions with brackets that cancel.
Multiply with linear factorisation
\[ \frac{2x+4}{5} \times \frac{15}{x+2} \]
Factorise first, then multiply and simplify.
Multiply with quadratic factorisation
\[ \frac{x^2-4}{x+3} \times \frac{x+3}{x-2} \]
Factorise a quadratic, then multiply and simplify.
Dividing Algebraic Fractions
Divide with single terms
\[ \frac{4x^2}{5} \div \frac{2x}{15} \]
Divide fractions with single terms by flipping.
Divide with factorised expressions
\[ \frac{x-1}{4} \div \frac{x-1}{8} \]
Divide fractions with brackets by flipping.
Divide with linear factorisation
\[ \frac{3x-6}{4} \div \frac{x-2}{8} \]
Factorise, flip, and simplify.
Divide with quadratic factorisation
\[ \frac{x^2+3x+2}{x-4} \div \frac{x+1}{x-4} \]
Factorise quadratic, flip, and simplify.
Adding Algebraic Fractions
Add with integer denominators (single terms)
\[ \frac{x}{2} + \frac{3x}{4} \]
Add fractions with integer denominators.
Add with integer denominators (expressions)
\[ \frac{x+1}{3} + \frac{2x-1}{6} \]
Add fractions with linear expression numerators.
Add with algebraic denominators
\[ \frac{3}{x} + \frac{5}{2x} \]
Add fractions with single-term algebraic denominators.
Add with linear denominators
\[ \frac{2}{x+1} + \frac{3}{x+2} \]
Add fractions with different linear denominators.
Add where one denominator is a factor
\[ \frac{3}{x+2} + \frac{5}{(x+2)^2} \]
Add when one denominator divides the other.
Subtracting Algebraic Fractions
Subtract with integer denominators (single terms)
\[ \frac{5x}{6} – \frac{x}{4} \]
Subtract fractions with integer denominators.
Subtract with integer denominators (expressions)
\[ \frac{3x+2}{4} – \frac{x+3}{2} \]
Subtract fractions with linear numerators.
Subtract with algebraic denominators
\[ \frac{7}{2x} – \frac{3}{x} \]
Subtract fractions with single-term algebraic denominators.
Subtract with linear denominators
\[ \frac{5}{x+3} – \frac{2}{x-1} \]
Subtract fractions with different linear denominators.
Subtract where one denominator is a factor
\[ \frac{4}{x-1} – \frac{2}{3(x-1)} \]
Subtract when one denominator divides the other.
Extensions
Add/subtract with factorisation
\[ \frac{3}{x+2} + \frac{1}{x^2+2x} \]
Factorise one denominator first.
Add/subtract with both factorisations
\[ \frac{2}{x^2-1} + \frac{3}{x^2+2x+1} \]
Factorise both denominators first.
Combined operations
\[ \frac{x+2}{x-1} \times \frac{x-1}{x+3} + \frac{1}{x+3} \]
Multiple operations with algebraic fractions.
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