# Algebraic fractions A algebraic expression it is a combination of variables, numbers and signs of operations (addition, subtraction, multiplication, division, empowerment, settlement, logarithms). In a algebraic fraction, the numerator and denominator are algebraic expressions. It has the same properties as a numerical fraction.

## Definition

An algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. For example: 2x / (2 + √x) is a fraction whose numerator is a monomial and the denominator a binomial.

## Simplification

For simplify or reduce a algebraic fraction we must convert it into an equivalent fraction whose terms are prime to each other. When this occurs, the fraction is irreducible and then it is in its simplest expression or to your minimal expression.

### How to simplify an algebraic fraction?

We divide the numerator and denominator by their common factors until they are prime to each other.

Example: 4x / (16x2Y)

We divide the numerator and denominator by 4x:

(4x ÷ 4x) / (16x2y ÷ 4x) = (1 ∙ 1) / (4 ∙ x ∙ y) = 1 / 4xy

Since 1 and 4xy have no factor in common, this fraction is irreducible.

### If the fractions have the same denominator

1. We simplify the given fractions if possible.
2. We add or subtract the numerators of the fractions and the common denominator remains.
3. We simplify the fraction that results, if possible.

Example: Add the following fractions: (√x + 1) / 9; (2 + x) / 9

Both fractions have the same denominator, therefore, we add the numerators and keep the common denominator:

[(√x + 1) / 9] + [(2 + x) / 9] = (√x + x + 3) / 9

### If the fractions have a different denominator

1. We simplify the given fractions if possible.
2. We reduce the fractions given to the lowest common denominator.
3. We add or subtract the numerators of the resulting fractions and keep the denominator obtained in step 2.
4. We simplify the fraction that results, if possible.

Example: subtract the following fractions: (x + 12) / 4; 3 / (x + 1)

Both fractions have different denominators, we must reduce them to the lowest common denominator. To do this, we find the lcm of the denominators: lcm = 4 (x + 1). We divide the lcm by the denominators of the fractions:

4 (x + 1) ÷ 4 = (x + 1) and 4 (x + 1) ÷ (x + 1) = 4

The quotients obtained are multiplied by the respective numerators, that is:

[(x + 12) / 4] - [3 / (x + 1)] = [(x + 12) (x + 1)] / [4 (x + 1)] - [(3 · 4) / 4 (x + 1)] =

= [(x2 + 13x + 12) / 4 (x + 1)] - [12/4 (x + 1)] =

We subtract the numerators

[(x2 + 13x + 12) / 4 (x + 1)] - [12/4 (x + 1)] = (x2 + 13x) / 4 (x + 1)

## Multiplication and division

### Rules for multiplying algebraic fractions

1. We factorize (as much as possible) the terms of the fractions we want to multiply.
2. We simplify, removing the common factors in the numerators and denominators.
3. We multiply the numerators together and the result is written as the numerator of the resulting fraction.
4. We multiply the denominators together and the result is written as the denominator of the resulting fraction.
5. We simplify the resulting fraction, if possible.

Example: multiply the following fractions: 25x / (5x + 5); 16x / (4x2)

[25x / (5x + 5)] × [16x / (4x2)] = {52x / [5 (x + 1)]} × [42x / (4x2)] = [5x / (x + 1)] × (4 / x) =

= [(5x) (4)] / [(x + 1) (x)] = 20x / [x (x + 1)]

We simplify the fraction:

20x / [x (x + 1)] = 20 / (x + 1)

### Rules for dividing algebraic fractions

1. We factorize (as much as possible) the terms of the fractions we want to divide.
2. We simplify, removing the common factors in the numerators and denominators.
3. We multiply the first fraction by the reciprocal of the second.
4. We simplify the resulting fraction, if possible.

Example: divide the following fractions: 7x / (√49x + 7); 3 / (√x + 1)

The reciprocal of the second fraction is:

(√x + 1) / 3

So:

[7x / (√49x + 7)] ÷ [3 / (√x + 1)] = [7x / (√49x + 7)] × [(√x + 1) / 3] =

= [(7 ∙ x) / (√72x + 1)] × [(√x + 1) / 3] = [(7 ∙ x) / 7 (√x + 1)] × [(√x + 1) / 3] =

= [x / (√x + 1)] × [(√x + 1) / 3] =

= [x (√x + 1)] / [3 (√x + 1)] =

= x / 3