# Reducing Rational Expressions

Rational expression can be defined as a fraction in which both the denominator and numerator are polynomials.
A rational expression can be reduced to its lowest term by cancelling any common terms that is present in both numerator and denominator. This is the same concept that we follow in ordinary fractions.
For example $\frac{21}{35}$ is not in its reduced form.
Common factor to both $21$ and $35$ is $7$
So cancelling both $7$ ,
$\frac{21}{35}=\frac{3\times7}{5\times7}$
=$\frac{3}{5}$
So, $\frac{3}{5}$ is the reduced form of $\frac{21}{35}$.
Same process is used for rational expressions,
For example
$\frac{(x-2)(x+4)}{x-2}$ is not the reduced form as there is $x-2$ common to both numerator and denominator.
Cancelling both we get the reduced form as ${(x+4)}$
It must be noted that while reducing a rational expression, only common factors can be cancelled and not common terms.For example, $\frac{25-5x}{25}$.
Here as there is a 25 common to both $25-5x$ and $25$, there will be a tendency to cancel them, but

that process is wrong.
First find factors of numerator and denominator separately
$25-5x=5(5-x)$ and $25=5\times5$
,So, $\frac{25-5x}{25}= \frac{5(5-x)}{5\times5}$
Here $5$ is the common factor, hence cancelling it out,
=$\frac{5-x}{5}$
This is the proper way of reducing a rational expression.
Lets consider an example of simplifying and reducing a rational expression to its lowest form,
$\frac{1+\frac{a}{b}}{1-\frac{a}{b}}$
Here, first take numerator and denominator separately.
${1+\frac{a}{b}}$
The highest common factor of $1$ and $b$ is $b$
So, ${1+\frac{a}{b}}=\frac{b}{b}+\frac{a}{b}$
=$\frac{b+a}{b}$
Next,${1-\frac{a}{b}}$
The highest common factor of $1$ and $b$ is $b$
So, ${1-\frac{a}{b}}=\frac{b}{b}-\frac{a}{b}$
=$\frac{b-a}{b}$
Hence,
$\frac{1+\frac{a}{b}}{1-\frac{a}{b}}= \frac{\frac{b+a}{b}}{\frac{b-a}{b}}$
=$\frac{b+a}{b}\times \frac{b}{b-a}$
Cancelling common term
=$\frac{b+a}{b-a}$