Algebraic fraction

An algebraic fraction is a special kind of fraction used in algebra. Unlike regular numbers, the top part (numerator) and bottom part (denominator) of an algebraic fraction are made from algebraic expressions. These expressions can include letters, like x, that stand for unknown numbers, along with operations such as addition, subtraction, and multiplication. For example, one algebraic fraction might look like (\frac{3x}{x^{2}+2x-3}), where both the top and bottom include the letter x. Just like regular fractions, algebraic fractions follow the same basic rules.

A rational fraction is a type of algebraic fraction where both the numerator and denominator are polynomials. Polynomials are expressions made by adding together terms of powers of x, like (x^2 + 2x - 3). So, a fraction such as (\frac{3x}{x^{2}+2x-3}) is a rational fraction because both the top and bottom are polynomials. However, not all algebraic fractions are rational. For instance, (\frac{\sqrt{x+2}}{x^{2}-3}) is an algebraic fraction, but it is not rational because the numerator includes a square root, which is not a polynomial. Understanding algebraic fractions helps solve more complex problems in algebra and higher math.

Terminology

In an algebraic fraction like a ÷ b, the top number a is called the numerator, and the bottom number b is called the denominator. Both the numerator and denominator are known as the terms of the fraction.

A complex fraction has a fraction in its numerator or denominator, while a simple fraction has no fractions inside it. When a fraction’s numerator and denominator share no common factors except 1, it is said to be in lowest terms. Numbers that aren’t written as fractions are called integral expressions, and they can always be turned into fractions by using 1 as the denominator. A mixed expression combines whole numbers with fractions.

Rational fractions

See also: Rational function

When the expressions in the top and bottom parts of a fraction are special math expressions called polynomials, the fraction is called a rational fraction. Rational fractions follow the same rules as regular fractions you already know.

A rational fraction is called "proper" if the top part is simpler than the bottom part. For example, 2x/(x²−1) is proper. If the top part is more complex, it's called "improper." Any improper rational fraction can be split into a simpler part and a proper fraction. For instance, (x³+x²+1)/(x²−5x+6) can be written as (x+6) + (24x−35)/(x²−5x+6).

Breaking a proper rational fraction into the sum of two or more simpler fractions is called resolving it into partial fractions. For example, 2x/(x²−1) can be written as 1/(x−1) + 1/(x+1). These simpler parts are called partial fractions.

Irrational fractions

An irrational fraction is a special type of fraction that includes a variable raised to a fractional exponent, like a square root or cube root in the exponent. For example, one such fraction looks like this:

[ \frac{x^{1/2} - \frac{1}{3}a}{x^{1/3} - x^{1/2}} ]

To make these fractions easier to work with, mathematicians use a process called rationalization. This means changing the fraction so that it no longer has fractional exponents. One way to do this is by finding the least common multiple of the exponents and then substituting the variable with another variable raised to that multiple. This turns the irrational fraction into a rational one, which is easier to handle in calculations.