# Polynomial equation of degree (Higher-degree polynomial equations)

A polynomial equation of degree \(n\) is an equation of the form \( a_n x^n+a_{n-1} x^{n-1}+⋯+a_1 x+a_0=0 \) where \(a_n, a_{n-1}, …, a_1 ,a_0 \) are constants and \(n\) is a non-negative integer. Here, \(x\) is the variable.

The degree of a polynomial equation is the highest power of the variable x in the equation.

For example, the equation \(2x^3–5x^2+3x–7=0\) is a polynomial equation of degree 3, because the highest power of \(x\) is 3. The constant term \(a_0\) is also included in the sum, even though it does not involve \(x\), to make it clear that this is a polynomial equation.

The solutions of a polynomial equation of degree \(n\) are called roots or zeros of the polynomial. The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) complex roots, counting multiplicities. This means that there are exactly \(n\) solutions to the equation \( a_n x^n +a_{n-1} x^{n-1} +⋯+a_1 x+ a_0 = 0 \), when we allow complex values of \(x\).

There are several techniques for finding the roots of polynomial equations of degree \(n\). For example, if \(n=1\), then the equation \(ax+b=0\) has a single root, which is \(x=- \frac{b}{a} \). For quadratic equations of the form \( ax^2 +bx+c=0 \), we can use the quadratic formula to find the roots. For cubic equations of the form \(ax^3+bx^2+cx+d=0 \), there is also a formula called **Cardano's formula**, which gives the roots in terms of the coefficients \(a\), \(b\), \(c\), \(d\). However, this formula can be quite complicated to use in practice. For polynomial equations of degree greater than 3, there is no general formula that gives the roots in terms of the coefficients, although there are some special cases where the roots can be found using other methods.

In general, finding the roots of a polynomial equation of degree n involves factoring the polynomial into linear and quadratic factors, and then solving each of the factors separately.

For example, consider the polynomial equation \(x^3–3x^2+2x=0\). We can factor this as \(x(x-1)(x-2)=0\), which gives us three solutions: \(x=0\), \(x=1\) and \(x=2\). These are the roots of the polynomial equation.

**Higher-degree polynomial equations** are equations of degree 4 or higher, such as quartic equations \( (ax^4+bx^3+cx^2+dx+e=0) \) and quintic equations \( (ax^5+bx^4+cx^3+dx^2+ex+f=0) \). These equations can be more difficult to solve than lower-degree polynomial equations and may require more advanced techniques, such as the use of complex numbers and group theory.

One important result in the theory of polynomial equations is the fundamental theorem of algebra, which states that every non-constant polynomial equation with complex coefficients has at least one complex root. This means that even if a polynomial equation cannot be solved using real numbers, it can always be solved using complex numbers.

There are several techniques for solving higher-degree polynomial equations. One approach is to try to factor the equation into simpler polynomials.

For example, the quartic equation \( x^4–5x^2+4=0 \) can be factored as \((x^2–4)(x^2–1)=0 \), which has roots \( x = \pm 1 \) and \( x = \pm 2 \). However, not all higher-degree polynomial equations can be factored in this way.

Another approach is to use numerical methods to approximate the roots of the equation. For example, the Newton-Raphson method can be used to iteratively find better and better approximations to the roots. However, these methods do not guarantee an exact solution and may require a large number of iterations to converge.

For quintic and higher-degree polynomial equations, there is no general formula for finding the roots in terms of the coefficients, as there are for quadratic, cubic, and quartic equations. This was proved by the mathematician Évariste Galois in the 19th century, using the theory of group theory. Instead, specific equations can be solved using specialized techniques or approximated using numerical methods.