Quartic equation (fourth-degree polynomial)

A quartic equation is a polynomial equation of degree four. This article explains its general form, key properties, historical solution methods, applications, and notable distinctions from lower- and higher-degree polynomials.

Author: Leandro Alegsa Created: November 13, 2022 Updated: June 8, 2026

A quartic equation is a polynomial equation of degree four, typically written as ax^4 + bx^3 + cx^2 + dx + e = 0 with a ≠ 0. In elementary algebra this form is the standard way to present a fourth-degree polynomial; its four roots (counting multiplicity) may be real or complex. The quartic is the highest-degree polynomial for which a general solution by radicals exists.

ax^{4}+bx^{3}+cx^{2}+dx+e

Form and key characteristics

Writing the polynomial as ax^4+bx^3+cx^2+dx+e clarifies coefficients and symmetry. A common simplification is the depressed quartic, obtained by substituting x = y - b/(4a) to remove the cubic term. The nature of the roots depends on the discriminant and factorization: a quartic can factor into two quadratics or into a linear times a cubic over some field.

Historical development and solution

The general algebraic solution for quartic equations was found in the 16th century by Lodovico Ferrari, building on methods used to solve the cubic. Ferrari introduced a technique that reduces the quartic to a resolvent cubic; solving that cubic yields parameters that allow the quartic to be written as a product of two quadratics and solved by square roots.

Uses and examples

Quartic equations appear in geometry (quartic curves), physics (certain potentials and optics problems), and engineering when modeling systems with fourth-power terms. Practical approaches often use numeric root-finding, symbolic factorization, or reduction to quadratics when coefficients permit.

Notable facts and distinctions

The quartic is the last degree with a general solution by radicals; quintic and higher degrees are not solvable in general.
Solving a quartic often involves solving an associated cubic (the resolvent cubic).
Galois theory explains why quartics are solvable: their symmetry groups fall into classes that permit radical expressions.

For more on polynomial theory and related background see general references on polynomial equations and algebraic methods.

Properties of quartic functions

In the following, let $f\colon \R\to \R$ a quartic function defined $a\not =0$ by $f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,$ with

Behaviour at infinity

As with all integer functions of even degree, the following applies

$\lim \limits _{{x\to +\infty }}f(x)=+\infty$ , $\lim \limits _{{x\to -\infty }}f(x)=+\infty$ ,

if the leading coefficient is positive, and

$\lim \limits _{{x\to +\infty }}f(x)=-\infty$ , $\lim \limits _{{x\to -\infty }}f(x)=-\infty$ ,

if is negative.

Nulls

A fourth degree polynomial has at most four zeros, but can also have no real zeros. If zeros are counted according to their multiplicity, it has exactly four complex zeros. If all zeros are real, the discriminant is nonnegative. The converse does not hold, the polynomial $x^{4}+4$ has positive discriminant but no real zeros.

For the (complex) zeros there is a solution formula, see quartic equation. The numerical finding of real zeros is possible, for example, with Newton's method.

Local extremes

As a polynomial function, differentiable arbitrarily many times; for its 1st derivative , we get the cubic function

$f'(x)=4ax^{3}+3bx^{2}+2cx+d$ .

If its discriminant is positive, then has exactly three local extrema, namely for a>0 one local maximum and two local minima or for a<0 two local maxima and one local minimum.

Turning Points

A quartic function has at most two inflection points $(x_{W};f(x_{W}))$ . The inflection points $x_{W}$ are the zeros of the 2nd derivative $f''(x)=12ax^{2}+6bx+2c$ .

Fourth degree polynomials

Let be any ring. Fourth degree polynomials over are expressions of the form

$ax^{4}+bx^{3}+cx^{2}+dx+e\in R[x]$

with $a,b,c,d,e\in R$ and $a\not =0$ . Formally, these are elements of the polynomial ring of degree 4, they define mappings from to . For $R=\mathbb {R}$ are quartic functions in the above sense.

If is an algebraically closed body, every fourth degree polynomial decays as a product of four linear factors.

More generally, quartic polynomials in variables are expressions of the form

$\sum _{i,j,k,l=1}^{n}a_{i,j,k,l}x_{i}x_{j}x_{k}x_{l}+\sum _{i,j,k=1}^{n}b_{i,j,k}x_{i}x_{j}x_{k}+\sum _{i,j=1}^{n}c_{i,j}x_{i}x_{j}+\sum _{i=1}^{n}d_{i}x_{i}+e\in R[x_{1},\ldots ,x_{n}]$ ,

Where not all $a_{i,j,k,l}$ are said to be zero. These polynomials define mappings from $R^{n}$ to . Their sets of zeros in $R^{n}$ are called quartic curves for n=2 and quartic surfaces for . n=3

Author

AlegsaOnline.com Quartic equation (fourth-degree polynomial) Leandro Alegsa

URL: https://en.alegsaonline.com/art/80409

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Alegsa, L. (June 8, 2026). Quartic equation (fourth-degree polynomial). AlegsaOnline.com. https://en.alegsaonline.com/art/80409

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BibTeX

@misc{alegsaonline_80409,
  author = {Alegsa, Leandro},
  title = {Quartic equation (fourth-degree polynomial)},
  year = {2026},
  howpublished = {AlegsaOnline.com},
  url = {https://en.alegsaonline.com/art/80409},
  note = {Updated: June 8, 2026; Language: en}
}

TXT

Leandro Alegsa. “Quartic equation (fourth-degree polynomial).” AlegsaOnline.com. Updated: June 8, 2026. https://en.alegsaonline.com/art/80409