Algebraic fractions Algebraic fractions are like normal fractions, but they contain algebraic expressions as the numerator and/or denominator. 3x 4x2 2a 3a + 2 and are two examples of algebraic fractions. The rules that apply to numerical fractions also apply to algebraic fractions. For example, if we multiply or divide the numerator and the denominator of a fraction by the same number or term, we produce an equivalent fraction. Teacher notes It is important to realize that, like numerical fractions, multiplying or dividing the numerator and the denominator of an algebraic fraction by the same number, term or expression does not change the value of the fraction. This fact is used both when simplifying algebraic fractions and when writing algebraic fractions over a common denominator to add or subtract them. 3x 4x2 3 4x 6 8x 3y 4xy 3(a + 2) 4x(a + 2) = = = =

Equivalent algebraic fractions Teacher notes It is important to realize that, like numerical fractions, multiplying or dividing the numerator and the denominator of an algebraic fraction by the same number, term or expression does not change the value of the fraction. This fact is used both when simplifying algebraic fractions and when writing algebraic fractions over a common denominator to add or subtract them.

Simplifying algebraic fractions Just like normal fractions, it is possible to simplify or cancel algebraic fractions. This is done using the same method of dividing the numerator and the denominator by common factors. 6ab 3ab2 How would you simplify the fraction ? 2 6ab 3ab2 = 6 × a × b 3 × a × b × b = 2 b

Simplifying algebraic fractions

Simplifying with brackets When dealing with brackets, it is important that all information has been dealt with before attempting any simplification. 5a2(x + y) 10a(x + y)2 How would you simplify the fraction ? 5a2(x + y) 10a(x + y)2 5 × a × a × (x + y) 10 × a × (x + y) × (x + y) = 2 a 2(x + y) =

Simplifying by factorizing Sometimes there is a need to factorize the numerator and the denominator before simplifying an algebraic fraction. 2a + a2 8 + 4a How would you simplify the fraction ? 2a + a2 8 + 4a a (2 + a) 4(2 + a) a 4 = = 6x + x2 12 + 6x How would you simplify ? 6x + x2 12 + 6x x (6 + x) 2(6 + x) x 2 = =

Spotting a pattern Algebraic fractions may appear that contain information that we know how to easily factorize. b2 – 36 3b – 18 How would you simplify the fraction ? b2 – 36 is the difference between two squares. b2 – 36 3b – 18 (b + 6)(b – 6) 3(b – 6) b + 6 3 = = Teacher notes Pupils should be encouraged to spot the difference between two squares whenever possible. If required, this can also be written as: b + 6 3 b 3 6 3 b 3 = + = + 2