Algebraic equation

In mathematics, an algebraic equation is a special kind of problem that helps us find unknown values. It looks like this: P = 0, where P is a polynomial. Polynomials are expressions made of numbers and letters (called variables). They are combined using addition, subtraction, and multiplication.

For example, an equation like ( x^{5} - 3x + 1 = 0 ) is an algebraic equation.

Algebraic equations can have one variable, like the example above, or many variables, such as ( y^{4} + \frac{xy}{2} - \frac{x^{3}}{3} + xy^{2} + y^{2} + \frac{1}{7} = 0 ). When there is only one variable, it is called a univariate equation. When there are multiple variables, it is called a multivariate equation.

Some algebraic equations can be solved easily using basic algebra. This works well for simple equations. For more complex equations, scientists and mathematicians use special methods to find answers that are very close to the correct ones.

Terminology

The term "algebraic equation" started when algebra mostly dealt with solving equations with one variable. In the 1800s, mathematicians made important discoveries that solved this problem.

Since then, algebra has grown to include many more types of equations, like those with roots and other complex expressions. Because of this, the term "algebraic equation" can sometimes be confusing. To avoid confusion, especially with equations that have more than one variable, many people now use the term "polynomial equation" instead.

Main article: Fundamental theorem of algebra
Main articles: Abel–Ruffini theorem, Galois theory

History

The study of algebraic equations started a very long time ago. Early Babylonian mathematicians around 2000 BC could solve some quadratic equations. In a quadratic equation, the highest power of the variable is 2.

Many mathematicians helped solve these equations over the years.

Important moments include:

• Indian mathematician Brahmagupta in the 7th century AD described the quadratic formula.
• In the 9th century, Muslim mathematicians like Muhammad ibn Musa al-Khwarizmi found the general solution for quadratic equations.
• During the Renaissance, Gerolamo Cardano shared solutions for equations of degree 3 and 4.
• In 1824, Niels Henrik Abel proved that equations of degree 5 and higher cannot always be solved using simple root expressions.

Areas of study

Algebraic equations are important in many parts of mathematics. Algebraic number theory studies equations that use rational numbers. Galois theory helps us know when these equations can be solved with special math tools. Algebraic geometry looks at the answers to equations with many variables.

Some equations can be changed so their numbers are whole numbers instead of fractions. For example, an equation with fractions can be turned into one with only whole numbers by multiplying by the right number. Not all equations are algebraic—some use functions like sine or exponentials, which are not polynomials.

Theory

Main article: Polynomial § Solving polynomial equations

An algebraic equation is a math problem where we try to find a number that makes a certain expression equal to zero. For example, in the equation x<sup>5</sup> − 3x + 1 = 0, we are looking for a value of x that makes this true. These equations can have one or more letters, called variables, and each letter can have different values.

A special rule says that a polynomial equation of degree n can have up to n solutions. This means if the highest power of x in the equation is 5, there can be at most 5 different values of x that solve it. Sometimes the solutions include numbers that are not real, called complex numbers.

Explicit solution of numerical equations

Main article: Quadratic equation

Main articles: Abel–Ruffini theorem and Galois group

Solving equations that look like ( P = 0 ), where ( P ) is a polynomial, means finding the values that make the equation true. For simple equations with one variable, like ( x^5 - 3x + 1 = 0 ), we try to break the polynomial into smaller parts. This is like factoring a number into smaller numbers that multiply together.

For quadratic equations, such as ( ax^2 + bx + c = 0 ), we use a special number called the discriminant ( \Delta = b^2 - 4ac ). If ( \Delta ) is positive, there are two solutions; if it is zero, there is one solution; and if it is negative, there are no real solutions. For more complex equations, there are special methods to solve them, but some equations need other approaches.