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E2.3 Algebraic fractions

Syllabus
0580–2028–2029
Topic
E2.3
Level
Extended

Add, subtract, multiply and divide algebraic fractions

Algebraic fractions follow the same operation rules as numerical fractions. Denominators must be non-zero, and the final expression should be simplified without changing its value.

Operation Reliable move
add or subtract use a lowest common denominator, rewrite every numerator, then combine
multiply factor first, multiply numerators and denominators, then cancel common factors
divide multiply by the reciprocal of the second fraction, then simplify

rac{2}{x-1}+ rac{3}{x+2}= rac{2(x+2)+3(x-1)}{(x-1)(x+2)}= rac{5x+1}{(x-1)(x+2)}

The common denominator must contain every required factor. In the example, $x
e1,-2$ because those values make an original denominator zero.

For multiplication, rac{4a}{5} imes rac{15}{8a}= rac32 for $a
e0.Fordivision,. For division, rac{3p}{7}\div rac{9p}{14q}= rac{3p}{7} imes rac{14q}{9p}= rac{2q}{3},with, withp
e0andandq
e0$.

Cancel only common factors in a product. Terms joined by ++ or −- cannot be cancelled: in racx+3xrac{x+3}{x}, the xx is not a factor of the whole numerator.

Factorise and simplify rational expressions

A rational expression simplifies when its numerator and denominator are written as products and a factor common to both is cancelled. The cancelled factor must be non-zero.

Factorise the numerator fully; factorise the denominator fully; identify identical factors; cancel only those factors; state every value excluded by the original denominator; expand the remaining factors only if a different final form is required.

rac{2x^2-5x-12}{3x^2-12x}= rac{(2x+3)(x-4)}{3x(x-4)}= rac{2x+3}{3x}

The original denominator is 3x(x−4)3x(x-4), so $x
e0,4.Although. Although(x-4)disappearsfromthesimplifiedexpression,disappears from the simplified expression,x=4$ is still excluded because it made the original expression undefined.

Check by multiplying the simplified numerator and denominator by the cancelled factor: rac{2x+3}{3x} imes rac{x-4}{x-4} reconstructs the factorised original expression whenever $x
e4$.

Cancellation removes factors, not matching-looking terms. For example, racx+5xrac{x+5}{x} cannot be reduced, while racx(x+5)x=x+5rac{x(x+5)}{x}=x+5 is valid only for $x
e0$.