I this chapter we will learn about the concept of algebraic fractions with examples.

To understand the chapter you should have basic idea about fractions and algebraic expressions.

## What are algebraic fractions ?

The fraction that contains **algebraic expression on numerator or denominator or both **are called **algebraic fractions**.

**How algebraic fractions are different from fraction numbers?**

Fraction number **contains integers or constant values on both numerator and denominators**.

While** algebraic fractions contain polynomials** on either numerator or denominator.

The polynomials in the algebraic expression contains both constants and variables.

**Note:**

Remember that in fractions the denominator cannot be equal to 0 otherwise the number would get ” Not defined “.

## Examples of algebraic fractions

Given below are examples of algebraic fractions with explanation.

\mathtt{( i) \ \frac{5}{x+3}}

Numerator contains number ⟹ 5

Denominator contains polynomial ⟹ ( x +3 )

\mathtt{( ii) \ \frac{x^{2} +4x+6}{-11}}

Numerator contains polynomial ⟹ \mathtt{x^{2} +4x+6}

Denominator contains integer ⟹ -11

\mathtt{(iii) \ \frac{\mathtt{x} +13}{x^{3} +6}}

Numerator contains polynomial ⟹ (x + 13)

Denominator contains polynomial ⟹ \mathtt{\left( x^{3} +6\right)}

\mathtt{( iv) \ \ \frac{x^{4} +12x^{2} +4}{1}}

Numerator contains polynomial ⟹ \mathtt{x^{4} +12x^{2} +4}

Denominator contains number ⟹ 1

\mathtt{( v) \ \ \frac{1-x^{2}}{1+x^{4}}}

Numerator contains polynomial ⟹ \mathtt{1-x^{2}}

Denominator contains polynomial ⟹ \mathtt{1+x^{4}}

### Can we have denominator 0 in algebraic fraction ?

No!!

In math, **division of any number with 0 is ” not defined “**. It means that we cannot get fixed value after division with 0.

So in algebraic fraction, **if denominator is polynomial then its value should not be equal to 0**.

For example, consider the below algebraic fraction;

\mathtt{\Longrightarrow \ \frac{5}{x-4}}

Here;

numerator contains number ⟹ 5

denominator contains polynomial ⟹ x – 4

Let value of x = 4;

Then fraction value becomes;

\mathtt{\Longrightarrow \ \frac{5}{4-4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{0}}\\\ \\ \mathtt{\Longrightarrow \ Not\ defined}

Hence, for fraction to be viable, the value of x cannot be equal to 4.

### Can we break algebraic fraction into small parts ?

Off course!!

If the numerator contains a polynomial then the polynomial can be broken down into smaller components keeping the same denominator.

**For example;**

(i) \mathtt{\frac{\mathtt{x^{2}} +3}{\mathtt{1+x}}}

Note that the numerator contains polynomial with two terms which are separated by addition sign.

Breakdown the individual terms and keep the same denominator.

\mathtt{\Longrightarrow \ \frac{x^{2}}{1+x} +\frac{3}{1+x}}

(ii) \mathtt{\frac{\mathtt{x^{3}} +2x-5}{\mathtt{x^{2} -7}}}

The numerator is a polynomial with three terms.

The fraction can be broken down as;

\mathtt{\Longrightarrow \ \frac{x^{3}}{x^{2} -7} +\frac{2x}{x^{2} -7} -\frac{5}{x^{2} -7}}

(iii) \mathtt{\frac{10}{5\mathtt{x^{2} +11x-3}}}

Here the numerator contain single constant number 10.

Hence, the **fraction cannot be broken down further**.