Algebraic Fractions

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Algebraic Fractions

Algebraic Fractions

Algebraic fraction or rational algebraic expression is a fraction whose numerator or denominator or both numerator and denominator are algebraic expressions. For example:




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Examples of algebraic fraction

Algebraic fraction should have non-zero denominator. If the denominator is zero (0) it is known as undefined or meaningless.

 

Basic Properties of Algebraic Fractions

The algebraic fraction whose denominator is not a zero has the following properties:

-        The value of fraction does not change if the numerator and denominator are multiplied by the same number.

Example: The value of fraction does not change if the numerator and denominator are multiplied by the same number.

-        The value of fraction does not change if the numerator and denominator are multiplied by the same polynomial.

Example: The value of fraction does not change if the numerator and denominator are multiplied by the same polynomial.

-        We can reduce the algebraic fractions by cancelling the common factors from numerator and denominator.

Example: We can reduce the algebraic fractions by cancelling the common factors from numerator and denominator.

The basic properties of the fraction help us to change algebraic fractions with different denominators into the fraction with same denominators. Such property of fractions makes easy to simplify the algebraic fractions.

 

Look at the following workout examples:

Example 1: For what value of x, are the following expressions undefined?

a) 6/x b) 7/(x-3) c) (x^2+8)/(2x-3)


Solution:

a) 6/x

An algebraic fraction is undefined when its denominator is 0. The denominator is x here.

             ∴ for x = 0, the given algebraic fraction is undefined.

 

b) 7/(x-3)

An algebraic fraction is undefined when its denominator is 0. The denominator is x - 3 here.

             ∴ for x - 3 = 0 or x = 3, it is undefined.

 

c) (x^2+8)/(2x-3)

An algebraic fraction is undefined when its denominator is 0. The denominator is 2x - 3 here.

             ∴ for 2x - 3 = 0 or 2x = 3 or x = 3/2, it is undefined.

Example 2: Reduce the following algebraic fractions into their lowest terms:

a. (6xy^2)/(2x^2 y) b. (a^2-b^2)/(ab-b^2 ) c. (x^2-1)/(x^2-2x+1) d. (x^2-16)/(x^2-9x+20)
 

Solution:

a) (6xy^2)/(2x^2 y) = (2 × 3 × x × y × y)/(2 × x × x × y) = 2y/x

 

b) (a^2-b^2)/(ab-b^2 ) = ((a+b)(a-b))/(b(a-b)) = (a+b)/b

 

c) (x^2-1)/(x^2-2x+1) = (x^2-1^2)/(x^2-2.x.1+1^2 ) = ((x+1)(x-1))/〖(x-1)〗^2 = ((x+1)(x-1))/((x-1)(x-1)) = (x+1)/(x-1)

 

d) (x^2-16)/(x^2-9x+20) = (x^2-4^2)/(x^2-(5+4)x+20) = ((x+4)(x-4))/(x^2-5x-4x+20) = ((x+4)(x-4))/(x(x-5)-4(x-5)) = ((x+4)(x-4))/((x-5)(x-4)) = (x+4)/(x-5)


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