# Polynomial fractions

A polynomial it is a combination of a finite set of variables, constants (fixed numbers called coefficients), and addition, subtraction, and multiplication arithmetic operations, as well as positive integer exponents. In a polynomial fraction, the numerator and denominator are polynomials. It has the same properties as a numerical fraction.

## Definition

A polynomial fraction is a fraction whose numerator and denominator are polynomials:

P (x) / Q (x) = polynomial / polynomial; with Q (x) ≠ 0

For example:

(x2 - Y2) / (x2+ 2xy + y2)

It is a polynomial fraction.

## Simplification

For simplify or reduce a polynomial 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 its minimum expression.

### How to simplify a polynomial fraction?

We factor polynomials as far as possible and remove factors common to the numerator and denominator.

Example: Simplify the following fraction: 3ab / (2a2x + 2a3)

• We factor the denominator:

3ab / (2a2x + 2a3) = 3ab / [2a2 (x + a)]

• We divide the numerator and denominator between a:

(3ab ÷ a) / [2a2(x + a) ÷ a] = 3b / [2a (x + a2)]

Thus:

3ab / (2a2x + 2a3) = 3b / [2a (x + a2)]

As 3b and 2a (x + a2) 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: (x2 + 2) / (x2 - one); x2 / (x2 - 1)

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

[(x2 + 2) / (x2 - 1)] + [x2 / (x2 - 1)] = (x2 + 2 + x2) / (x2 - 1) =

= 2 (x2 + 1) / (x2 - 1)

### 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 (x + 2)] / (x2 - one); x / (x - 1)

Both fractions have different denominators, we must reduce them to the lowest common denominator. For this, we find the lcm of the denominators: lcm = x2 - 1. We divide the lcm between the denominators of the fractions:

(x2 - 1) ÷ (x2 - 1) = 1 y (x2 - 1) ÷ (x - 1) = x + 1

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

{[x (x + 2)] / (x2 - 1)} - [x / (x - 1)] = {[x (x + 2)] / (x2 - 1)} - {[x (x - 1)] / (x2 - 1)} =

= [(x2 + 2x) / (x2 - 1)] - [(x2 - x) / (x2 – 1)] =

We subtract the numerators

= (x2 + 2x - x2 + x) / (x2 - 1) =

= (x2 + x) / (x2 – 1) =

= [x (x + 1)] / (x2 – 1)

Like x2 - 1 = (x + 1) (x - 1), we have

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

= x / (x - 1)

## Multiplication and division

### Rules for multiplying polynomial 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: (5x2 - 5) / (25x + 25); 16x / (4x2 - 4)

Like x2 - 1 = (x + 1) (x - 1)

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

= [(x - 1) / 5] × {4x / [(x - 1) (x + 1)]} =

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

We simplify the fraction:

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

### Rules for dividing polynomial 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: (x3 - x) / (2x2 + 6x); (5x2 - 5x) / (2x + 6).

[(x3 - x) / (2x2 + 6x)] ÷ [(5x2 - 5x) / (2x + 6)] = [(x3 - x) / (2x2 + 6x)] ÷ [(2x + 6) / (5x2 - 5x) (2x + 6)] =

= [x (x2 - 1) / 2x (x + 3)] × [2 (x + 3) / 5x (x - 1)] =

= [2 (x2 - 1) (x + 3)] / [10 (x + 3) (x - 1)] =

Like x2 - 1 = (x + 1) (x - 1)

= [2 (x + 1) (x - 1) (x + 3)] / [10 (x + 3) (x - 1)] =

= (x + 1) / 5