Algebraic Fractions – SS2 Mathematics Lesson Note (Second Term)

Simplifying algebraic fractions

To simplify an algebraic fraction means to reduce it to its lowest term.  This is done by factoring out the common factors in the numerator and the denominator.  When simplifying, remember, the following facts:

1. x² – y² = (x + y) (x – y) (difference of two squares)
2. (x + y)² = x² + 2xy + y²   (perfect squares)

(x – Y)² = x² – 2xy + y²

iii. x = -x

    -y    y¹

1. -m = -m

      n       n

1. x = m

    y    y

Simplify the following fractions:

1. a) 3x² + 9x²y²

         3x²y

= 3x²(1 + 3y²)   = 1 + 3y²

      3x² × y               y

1. b) x² – y² + 3x + 3y

          x – y + 3

= (x+y)(x-y) + 3(x + y) = x + y

             x – y + 3

1. c) 8y³ – 16y²

 2y³ – 2y² – 4y

= 8y²(y – 2)             =      48y²

 2y(y + 1)(y – 2)            2y(y + 1)

=   4y

    y + 1

   Operations in algebraic fractions

1. Simply  2     +          1

             3x + 3        2x + 4

= 2(2x + 4) + (3x + 3)

       (3x + 3)(2x + 4)

= 4x + 8 + 3x + 3

    (3x + 3)(2x + 4)

=    7x + 11

(3x + 3)(2x + 4)

2. x²-y²   × 2x²-xy-y² ÷ y-x

x²-2xy+y      xy+x²          x²-xy

Factorizes each term to get

= (x-y)(x+y)  × (x-y)(2x+y) ÷ y-x

   (x-y)(x-y)        x(y+x)          x(x-y)

Change ÷ to × and then convert to obtain 

= (x-y)(x+y)  × (x-y)(2x+y) × x(x-y)

   (x-y)(x-y)        x(y+x)             y-x

But x-y = -1(y-x)

(2x+y)(x-y) = (x-y)(2x+y)

     y-x                 -(x-y)

= -(2x + y) = -2x – y