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:

(x^{2} - Y^{2}) / (x^{2}+ 2xy + y^{2})

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 / (2a^{2}x + 2a^{3})

- We factor the denominator:

3ab / (2a^{2}x + 2a^{3}) = 3ab / [2a^{2} (x + a)]

- We divide the numerator and denominator between a:

(3ab ÷ a) / [2a^{2}(x + a) ÷ a] = 3b / [2a (x + a^{2})]

Thus:

3ab / (2a^{2}x + 2a^{3}) = 3b / [2a (x + a^{2})]

As 3b and 2a (x + a^{2}) have no factor in common, this fraction is irreducible.

**Add and subtract**

**If the fractions have the same denominator**

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

**Example**: Add the following fractions: (x^{2 }+ 2) / (x^{2} - one); x^{2 }/ (x^{2 }- 1)

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

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

= 2 (x^{2} + 1) / (x^{2 }- 1)

**If the fractions have a different denominator**

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

**Example**: subtract the following fractions: [x (x + 2)] / (x^{2 }- 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 = x^{2} - 1. We divide the lcm between the denominators of the fractions:

(x^{2 }- 1) ÷ (x^{2 }- 1) = 1 y (x^{2 }- 1) ÷ (x - 1) = x + 1

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

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

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

We subtract the numerators

= (x^{2 }+ 2x - x^{2 }+ x) / (x^{2 }- 1) =

= (x^{2 }+ x) / (x^{2 }– 1) =

= [x (x + 1)] / (x^{2 }– 1)

Like x^{2 }- 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**

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

**Example**: multiply the following fractions: (5x^{2 }- 5) / (25x + 25); 16x / (4x^{2 }- 4)

Like x^{2 }- 1 = (x + 1) (x - 1)

[(5x^{2 }- 5) / (25x + 25)] × [16x / (4x^{2 }- 4)] = {[5 (x - 1) (x + 1)] / [5^{2}(x + 1)]} × {(4^{2}x) / [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**

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

**Example**: divide the following fractions: (x^{3} - x) / (2x^{2 }+ 6x); (5x^{2 }- 5x) / (2x + 6).

[(x^{3} - x) / (2x^{2 }+ 6x)] ÷ [(5x^{2 }- 5x) / (2x + 6)] = [(x^{3} - x) / (2x^{2 }+ 6x)] ÷ [(2x + 6) / (5x^{2 }- 5x) (2x + 6)] =

= [x (x^{2 }- 1) / 2x (x + 3)] × [2 (x + 3) / 5x (x - 1)] =

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

Like x^{2 }- 1 = (x + 1) (x - 1)

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

= (x + 1) / 5