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**

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.

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.

Reduce the rational
expressions to its lowest form

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)

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

(i)

(ii)

(iii)

(i)

The expression is undefined when 8*x* = 0 or *x* = 0 . Hence the excluded
value is 0.

(ii)

The expression is undefined when 8*p* ^{2} + 13*p* + 5 = 0

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

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

(iii)

Here *x*^{2}
≥ 0 for all *x*. Therefore , *x*^{2} + 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*/(*x*^{2}+1).

Tags : Definition, Reduction, Example, Solution | Algebra Definition, Reduction, Example, Solution | Algebra

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