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# Rational Expressions

An expression is called a rational expression if it can be written in the form p(x) / q(x) where p(x) and q(x) are polynomials and q(x) ≠ 0 . A rational expression is the ratio q (x) of two polynomials.

Rational Expressions

## Definition :

An expression is called a rational expression if it can be written in the form p(x) / q(x) where p(x) and q(x) are polynomials and q(x) ≠ 0 . A rational expression is the ratio q (x) of two polynomials.

The following are examples of rational expressions. The rational expressions are applied for describing distance-time, modeling multi-task problems, to combine workers or machines to complete a job schedule and much more.

## 1. Reduction of Rational Expression

A rational expression p(x) / q(x) is said to be in its lowest form if GCD ( p(x ), q(x)) = 1.

To reduce a rational expression to its lowest form, follow the given steps

(i) Factorize the numerator and the denominator

(ii) If there are common factors in the numerator and denominator, cancel them.

(iii) The resulting expression will be a rational expression in its lowest form.

### Example 3.13

Reduce the rational expressions to its lowest form ### Solution ## 2. Excluded Value

A value that makes a rational expression (in its lowest form) undefined is called an Excluded value.

To find excluded value for a given rational expression in its lowest form, say p(x) / q(x), consider the denominator q(x) = 0.

For example, the rational expression 5/(x-10) is undefined when x = 10 . So, 10 is called an excluded value for 5/(x-10)

### Example 3.14

Find the excluded values of the following expressions (if any).

(i) (ii) (iii) ### Solution

(i) The expression is undefined when 8x = 0 or x = 0 . Hence the excluded value is 0.

(ii) The expression is undefined when 8p 2 + 13p + 5 = 0

that is, (8p + 5)(p + 1) = 0

P= −5/8,  p = −1 . The excluded values are -5/8 and -1.

(iii) Here x2  ≥ 0 for all x. Therefore , x2 + 1 ≥ 0 + 1 = 1, Hence,  x 2 + 1 ≠ 0 for any x. Therefore, Therefore, there can be no real excluded values for the given rational expression x/(x2+1).

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10th Mathematics : Algebra : Rational Expressions | Definition, Reduction, Example, Solution | Algebra

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