Algebraic fractions 3 x 2 a and are

Algebraic fractions 3 x 2 a and are examples of algebraic fractions. 2 4 x 3 a + 2 The rules that apply to numerical fractions also apply to algebraic fractions. For example, if we multiply or divide the numerator or the denominator of a fraction by the same number or term we produce an equivalent fraction. For example, 3 y 3(a + 2) 3 x 3 6 = = 2 4 x 4 x 8 x 4 xy 4 x(a + 2) 1 of 73 © Boardworks Ltd 2005

Equivalent algebraic fractions 2 of 73 © Boardworks Ltd 2005

Simplifying algebraic fractions We simplify or cancel algebraic fractions in the same way as numerical fractions, by dividing the numerator and the denominator by common factors. For example, 6 ab Simplify 3 ab 2 2 6 ab 6×a×b = 2 3 ab 3×a×b×b 2 = b 3 of 73 © Boardworks Ltd 2005

Simplifying algebraic fractions Sometimes we need to factorize the numerator and the denominator before we can simplify an algebraic fraction. For example, 2 a + a 2 Simplify 8 + 4 a 2 a + a 2 8 + 4 a a (2 + a) = 4(2 + a) a = 4 4 of 73 © Boardworks Ltd 2005

Simplifying algebraic fractions b 2 – 36 is the difference between two squares. b 2 – 36 Simplify 3 b – 18 (b + 6)(b – 6) b 2 – 36 = 3(b – 6) 3 b – 18 b+6 3 If required, we can write this as = b b 6 + = + 2 3 3 3 5 of 73 © Boardworks Ltd 2005

Manipulating algebraic fractions Remember, a fraction written in the form a+b c can be written as a b + c c However, a fraction written in the form c a+b cannot be written as c c + a b For example, 1+2 3 6 of 73 1 2 = + 3 3 but 3 1+2 3 3 = + 1 2 © Boardworks Ltd 2005

Multiplying and dividing algebraic fractions We can multiply and divide algebraic fractions using the same rules that we use for numerical fractions. In general, and, For example, a c ac × = b d bd a c a d ad ÷ = × = b d b c bc 3 6 p 3 p 3 p 2 = × = 2(1 – p) 4(1 – p) 4 2 7 of 73 © Boardworks Ltd 2005

Multiplying and dividing algebraic fractions 2 4 ? What is ÷ 3 y – 6 y– 2 2 3 y – 6 2 4 ÷ = y– 2 3 y – 6 This is the reciprocal 4 of y– 2 × 4 y– 2 2 = × 3(y – 2) 4 2 1 = 6 8 of 73 © Boardworks Ltd 2005

Adding algebraic fractions We can add algebraic fractions using the same method that we use for numerical fractions. For example, 1 2 What is + ? a b We need to write the fractions over a common denominator before we can add them. b 2 a b + 2 a 1 2 + = a b ab ab ab In general, a c ad + bc + = b d bd 9 of 73 © Boardworks Ltd 2005

Adding algebraic fractions y 3 What is + ? x 2 We need to write the fractions over a common denominator before we can add them. y y×x 3 3× 2 + = + x x× 2 2×x 2 xy 6 + = 2 x 2 x 6 + xy = 2 x 10 of 73 © Boardworks Ltd 2005

Subtracting algebraic fractions We can also subtract algebraic fractions using the same method as we use for numerical fractions. For example, p q What is – ? 3 2 We need to write the fractions over a common denominator before we can subtract them. p q 2 p 3 q 2 p – 3 q – = 3 2 6 6 6 In general, a c ad – bc – = b d bd 11 of 73 © Boardworks Ltd 2005

Subtracting algebraic fractions 2+p 3 What is – ? 2 q 4 2+p 3 (2 + p) × 2 q 3× 4 – = – 2 q 4 4 × 2 q 2 q × 4 2 q(2 + p) 12 = – 8 q 8 q 2 q(2 + p) – 12 = 8 q 4 6 q(2 + p) – 6 = 4 q 12 of 73 © Boardworks Ltd 2005

Addition pyramid – algebraic fractions 13 of 73 © Boardworks Ltd 2005