The division is one of the arithmetic operations. The division of algebraic fractions is similar to the division of numbers. Here, you need to factorize the numerators and denominators of the fractions. Cancel the like factors in a fraction and reverse the denominator fraction and multiply it with numerator fraction to get the division value. Find the solved example questions on the division of algebraic fractions.

How to Divide Algebraic Fractions?

Check the simple and easy steps to divide algebraic fractions in the following sections.

  • Take two algebraic fractions. One in the numerator and the second in the denominator.
  • Find the factors of fractions.
  • Get first fraction x 1 / second fraction.
  • Cancel the like terms in the numerator and denominator.
  • Multiply the numerators, denominators to get the result.

Solved Examples on Dividing Algebraic Fractions

Example 1.

Determine the quotient of the algebraic fractions: x / (x² – x) ÷ 10x / (x² + x – 2)?

Solution:

Given that,

x / (x² – x) ÷ 10x / (x² + x – 2)

Factorize the fractions and cancel the common terms.

x / x (x – 1) ÷ 10x / (x² + 2x – x – 2)

= x / x (x – 1) ÷ 10x / (x(x + 2) -1(x + 2))

= x / x (x – 1) ÷ 10x / (x + 2) (x – 1)

= 1/(x – 1) ÷ 10x / (x + 2) (x – 1)

= 1/(x – 1) x (x + 2) (x – 1) / 10x

= (x + 2) (x – 1) / 10x (x – 1)

= (x + 2) / 10x

Example 2.

Divide the algebraic fractions and express in the lowest form: 9x²+12x+4 / 4x²-27x-7 ÷ 12x²+5x-2 / 16x²-1?

Solution:

Given that,

9x²+12x+4 / 4x²-27x-7 ÷ 12x²+5x-2 / 16x²-1

Factorize the fractions and cancel the common terms.

= 9x²+6x+6x+4 / 4x²-28x+x-7 ÷12x²+8x-3x-2 / (4x)²-1²

= 3x(3x+2)+2(3x+2) / 4x(x-7)+1(x-7) ÷ 4x(3x+2)-1(3x+2) / (4x+1)(4x-1)

= (3x+2)(3x+2) / (x-7)(4x+1) ÷ (3x+2)(4x-1) / (4x+1)(4x-1)

= (3x+2)(3x+2) / (x-7)(4x+1) ÷ (3x+2) / (4x+1)

= (3x+2)(3x+2) / (x-7)(4x+1) * (4x+1) / (3x+2)

= (3x+2)(3x+2)(4x+1) / (x-7)(4x+1)(3x+2)

= 3x+2 / x-7

Example 3.

Find the quotient of the algebraic fractions:

x²+11x+24 / x²-15x+56 ÷ x²-x-12 / x²-11x+28

Solution:

Given that,

x²+11x+24 / x²-15x+56 ÷ x²-x-12 / x²-11x+28

Factorize the algebraic fractions and cancel the common terms.

= x²+8x+3x+24 / x²-8x-7x+56 ÷ x²-4x+3x-12 / x²-7x-4x+28

= x(x+8)+3(x+8) / x(x-8)-7(x-8) ÷ x(x-4)+3(x-4) / x(x-7)-4(x-7)

= (x+8)(x+3) / (x-8)(x-7) ÷ (x-4)(x+3) / (x-4)(x-7)

= (x+8)(x+3) / (x-8)(x-7) ÷ (x+3) / (x-4)

= (x+8)(x+3) / (x-8)(x-7) * (x-4) / (x+3)

= (x+8)(x+3)(x-4) / (x-8)(x-7)(x+3)

= (x+8)(x-4) / (x-8)(x-7)

= x²+4x-32 / x²-15x+56