Algebraic equation

Given an equation in unknown x

${\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}$,

with coefficients in a field K, one can equivalently say that the solutions of (E) in K are the roots in K of the polynomial

${\displaystyle P=a_{n}X^{n}+a_{n-1}X^{n-1}+\dots +a_{1}X+a_{0}\quad \in K[X]}$.

It can be shown that a polynomial of degree n in a field has at most n roots. The equation (E) therefore has at most n solutions.

If K' is a field extension of K, one may consider (E) to be an equation with coefficients in K and the solutions of (E) in K are also solutions in K' (the converse does not hold in general). It is always possible to find a field extension of K known as the rupture field of the polynomial P, in which (E) has at least one solution.

The fundamental theorem of algebra states that the field of the complex numbers is closed algebraically, that is, all polynomial equations with complex coefficients and degree at least one have a solution.

It follows that all polynomial equations of degree 1 or more with real coefficients have a complex solution. On the other hand, an equation such as ${\displaystyle x^{2}+1=0}$ does not have a solution in ${\displaystyle \mathbb {R} }$ (the solutions are the imaginary units i and –i).

While the real solutions of real equations are intuitive (they are the x-coordinates of the points where the curve y = P(x) intersects the x-axis), the existence of complex solutions to real equations can be surprising and less easy to visualize.

However, a monic polynomial of odd degree must necessarily have a real root. The associated polynomial function in x is continuous, and it approaches ${\displaystyle -\infty }$ as x approaches ${\displaystyle -\infty }$ and ${\displaystyle +\infty }$ as x approaches ${\displaystyle +\infty }$. By the intermediate value theorem, it must therefore assume the value zero at some real x, which is then a solution of the polynomial equation.

There exist formulas giving the solutions of real or complex polynomials of degree less than or equal to four as a function of their coefficients. Abel showed that it is not possible to find such a formula in general (using only the four arithmetic operations and taking roots) for equations of degree five or higher. Galois theory provides a criterion which allows one to determine whether the solution to a given polynomial equation can be expressed using radicals.

The explicit solution of a real or complex equation of degree 1 is trivial. Solving an equation of higher degree n reduces to factoring the associated polynomial, that is, rewriting (E) in the form

${\displaystyle a_{n}(x-z_{1})\dots (x-z_{n})=0}$,

where the solutions are then the ${\displaystyle z_{1},\dots ,z_{n}}$. The problem is then to express the ${\displaystyle z_{i}}$ in terms of the ${\displaystyle a_{i}}$.

This approach applies more generally if the coefficients and solutions belong to an integral domain.

If an equation P(x) = 0 of degree n has a rational root α, the associated polynomial can be factored to give the form P(X) = (X – α)Q(X) (by dividing P(X) by X – α or by writing P(X) – P(α) as a linear combination of terms of the form X^k – α^k, and factoring out X – α. Solving P(x) = 0 thus reduces to solving the degree n – 1 equation Q(x) = 0. See for example the case n = 3.

To solve an equation of degree n,

${\displaystyle (\mathrm {E} )\qquad a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}=0}$,

a common preliminary step is to eliminate the degree-n - 1 term: by setting ${\displaystyle x=y-{\frac {a_{n-1}}{n\,a_{n}}}}$, equation (E) becomes

${\displaystyle a_{n}y^{n}+b_{n-2}y^{n-2}+\dots +b_{1}x+b_{0}=0}$.

Leonhard Euler developed this technique for the case n = 3 but it is also applicable to the case n = 4, for example.

To solve a quadratic equation of the form ${\displaystyle ax^{2}+bx+c=0}$ one calculates the discriminant Δ defined by ${\displaystyle \Delta =b^{2}-4ac}$.

If the polynomial has real coefficients, it has:

• two distinct real roots if ${\displaystyle \Delta >0}$ ;
• one real double root if ${\displaystyle \Delta =0}$ ;
• no real root if ${\displaystyle \Delta <0}$, but two complex conjugate roots.

The best-known method for solving cubic equations, by writing roots in terms of radicals, is Cardano's formula.

For detailed discussions of some solution methods see:

• Tschirnhaus transformation (general method, not guaranteed to succeed);
• Bezout method (general method, not guaranteed to succeed);
• Ferrari method (solutions for degree 4);
• Euler method (solutions for degree 4);
• Lagrange method (solutions for degree 4);
• Descartes method (solutions for degree 2 or 4);

A quartic equation ${\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}$ with ${\displaystyle a\neq 0}$ may be reduced to a quadratic equation by a change of variable provided it is either biquadratic (b = d = 0) or quasi-palindromic (e = a, d = b).

Some cubic and quartic equations can be solved using trigonometry or hyperbolic functions.

Évariste Galois and Niels Henrik Abel showed independently that in general a polynomial of degree 5 or higher is not solvable using radicals. Some particular equations do have solutions, such as those associated with the cyclotomic polynomials of degrees 5 and 17.

Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable using elliptical functions.

Otherwise, one may find numerical approximations to the roots using root-finding algorithms, such as Newton's method.