## Introduction

- Recall that a
*rational fraction*is simply the ratio of two polynomials such as $\frac{{x}^{2}+2x+1}{2x-1}$, $\frac{2x}{1-{x}^{2}}$, $2{x}^{4}-3x+18$, etc. - A
*complex rational fraction*is a special rational fraction in which the dividend, divisor, or both contains rational fractions. - Since they appear to be fractions within a fraction, they are named complex rational fractions.
- Here are a few examples of complex rational fractions: $\frac{\left({\displaystyle \frac{2x}{{x}^{2}-4}}\right)}{\left({\displaystyle \frac{x-5}{{x}^{2}-2x+1}}\right)}$, $\frac{1}{\left({\displaystyle \frac{8x}{{x}^{2}-4}}\right)}$, $\frac{\left({\displaystyle \frac{{x}^{2}-4x+3}{x-1}}\right)}{8{x}^{3}}$, etc.
- It should be noted that we always exclude values that would make any denominator zero.

## Simplifying Complex Fractions

A complex fraction is said to be in its simplest form if it can't be further reduced, i.e., its numerator and denominator must not have any common factor other than 1.

We can simplify complex rational fractions in two ways, as described below.

**(1) Simplifying a complex rational fraction by writing it as division.**

The steps to be followed while using this method are as follows -

- Rewrite the complex rational fraction as division.
- Multiply the numerator by the reciprocal of the divisor.
- Factor both the numerators and denominators completely.
- Simply the expression (cancel all the common factors).

**(2) Simplifying a complex rational fraction using the least common denominator (LCD).**

It is an alternate method for simplifying complex rational fractions and involves the following steps -

- Find the LCD of all fractions in the complex rational fraction.
- Multiply the numerator and denominator by the obtained LCD.
- Factor both the numerators and denominators completely.
- Simply the expression (cancel all the common factors).

## Solved Examples

**Example 1:** Simplify $\frac{\left({\displaystyle \frac{8}{x+3}}\right)}{\left({\displaystyle \frac{64}{{x}^{2}-9}}\right)}$.

**Solution:** Simplify* the complex rational fraction* *by writing* *it as a division* to solve this problem.

$\frac{\left({\displaystyle \frac{8}{x+3}}\right)}{\left({\displaystyle \frac{64}{{x}^{2}-9}}\right)}$$=\left(\frac{8}{x+3}\right)\xf7\left(\frac{64}{{x}^{2}-9}\right)$$=\left(\frac{8}{x+3}\right)\times \left(\frac{{x}^{2}-9}{64}\right)$$=\frac{8}{\left(x+3\right)}\times \frac{\left(x+3\right)\left(x-3\right)}{64}$$=\frac{\left(x-3\right)}{8}$

**Example 2: **Simplify $\frac{\left({\displaystyle \frac{6}{{x}^{2}-6x+5}}\right)}{\left({\displaystyle \frac{3}{x-5}}\right)}$.

**Solution:** Simplify* the complex rational fraction* *by writing it as a division* to solve this problem.

$\frac{\left({\displaystyle \frac{6}{{x}^{2}-6x+5}}\right)}{\left({\displaystyle \frac{3}{x-5}}\right)}$$=\left(\frac{6}{{x}^{2}-6x+5}\right)\xf7\left(\frac{3}{x-5}\right)$$=\left(\frac{6}{{x}^{2}-6x+5}\right)\times \left(\frac{x-5}{3}\right)$$=\frac{6}{\left(x-5\right)\left(x-1\right)}\times \frac{\left(x-5\right)}{3}$$=\frac{2}{\left(x-1\right)}$

**Example 3: **Simplify $\frac{\left({\displaystyle 1-\frac{2}{x}}\right)}{\left({\displaystyle 1+\frac{5}{y}}\right)}$.

**Solution:** Simplify* the complex rational fraction* *by using LCD* to solve this problem. The LCD of all the given fractions is $xy$, multiply both the numerator and denominator by $xy$.

$\frac{\left({\displaystyle 1-\frac{2}{x}}\right)}{\left({\displaystyle 1+\frac{5}{y}}\right)}$$=\frac{\left({\displaystyle 1-\frac{2}{x}}\right)\times xy}{\left({\displaystyle 1+\frac{5}{y}}\right)\times xy}$$=\frac{{\displaystyle xy-2y}}{{\displaystyle xy+5x}}$

**Example 4: **Simplify $\frac{\left({\displaystyle \frac{{\displaystyle 1}}{5}+\frac{1}{x}}\right)}{\left({\displaystyle \frac{{\displaystyle 1}}{25}-\frac{1}{{x}^{2}}}\right)}$.

**Solution:** Simplify* the complex rational fraction* *by using LCD* to solve this problem. The LCD of all the given fractions is $25{x}^{2}$, multiply both the numerator and denominator by $25{x}^{2}$.

$\frac{\left({\displaystyle \frac{{\displaystyle 1}}{5}+\frac{1}{x}}\right)}{\left({\displaystyle \frac{{\displaystyle 1}}{25}-\frac{1}{{x}^{2}}}\right)}$$=\frac{\left({\displaystyle \frac{{\displaystyle 1}}{5}+\frac{1}{x}}\right)\times 25{x}^{2}}{\left({\displaystyle \frac{{\displaystyle 1}}{25}-\frac{1}{{x}^{2}}}\right)\times 25{x}^{2}}$$=\frac{{\displaystyle 5{x}^{2}+25x}}{{\displaystyle {x}^{2}-25}}$$=\frac{{\displaystyle 5x\left(x+5\right)}}{{\displaystyle \left(x+5\right)\left(x-5\right)}}$$=\frac{{\displaystyle 5x}}{{\displaystyle \left(x-5\right)}}$

## Cheat Sheet

- Complex rational fractions are quotients with rational fractions in the divisor, dividend, or both.
- Simplifying a complex rational fraction means reducing it into a form in which the numerator and denominator of the final result must not have any common factor other than 1.
- In simplifying
*a complex rational fraction by writing it as division*, we first express the given complex rational fraction as division, followed by multiplying the numerator by the divisor's reciprocal and simplifying it. - In simplifying
*a complex rational fraction by least common denominator (LCD),*we first multiply the numerator and denominator by the LCD of all given fractions and then simplify it.

## Blunder Areas

- Rational fractions are different from algebraic fractions.
- Rational fractions involve a ratio of polynomials, while in algebraic fractions, the numerator and denominator are algebraic expressions.
- All polynomials are algebraic expressions, but all algebraic expressions are not polynomials.

- Abhishek Tiwari
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