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Quartic function

Quartic function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Quartic function rather than just read about it. In short: In algebra, a quartic function is a function of the form f ( x ) = a x 4 + b x 3 + c x 2 + d x + e , {\displaystyle f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,} where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form a x 4 + b x 3 + c x 2 + d x + e = 0 , {\displays…

Quartic function — main illustration
Quartic function — illustration

Key takeaways

  • Quartic function belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Quartic function to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Quartic function from memory before moving on to harder problems.

Reference excerpt

In algebra, a quartic function is a function of the form

f ( x ) = a x 4 + b x 3 + c x 2 + d x + e , {\displaystyle f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,}

where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form

a x 4 + b x 3 + c x 2 + d x + e = 0 , {\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0,}

where a ≠ 0. The derivative of a quartic function is a cubic function. Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form

f ( x ) = a x 4 + c x 2 + e . {\displaystyle f(x)=ax^{4}+cx^{2}+e.}

Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum. The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals, according to the Abel–Ruffini theorem.

History Lodovico Ferrari is credited with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately. The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book Ars Magna. The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois prior to dying in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result.

Applications Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus. It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics. Here are examples of other geometric problems whose solution involves solving a quartic equation. In computer-aided manufacturing, the torus is a shape that is commonly associated with the endmill cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the z-axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated. A quartic equation arises also in the process of solving the crossed ladders problem, in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found. In optics, Alhazen's problem is "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer." This leads to a quartic equation. Finding the distance of closest approach of two ellipses involves solving a quartic equation. The eigenvalues of a 4×4 matrix are the roots of a quartic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a fourth-order linear difference equation or differential equation is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending. Intersections between spheres, cylinders, or other quadrics can be found using quartic equations.

Inflection points and golden ratio Letting F and G be the distinct inflection points of the graph of a quartic function, and letting H be the intersection of the inflection secant line FG and the quartic, nearer to G than to F, then G divides FH into the golden section:

F G G H = 1 + 5 2 = φ ( the golden ratio ) . {\displaystyle {\frac {FG}{GH}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \;({\text{the golden ratio}}).}

Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area.

Solution

Nature of the roots Given the general quartic equation

… excerpt ends here. Continue reading the full article.

Illustrations

Quartic function: Graph of a polynomial of degree 4, with 3 critical points and four real roots (crossings of the x axis) (and thus no complex roots). If one or the other of the local minima were above the x axis, or if the local maximum were below it, or if there were no local maximum and one minimum below the x axis, there would only be two real roots (and two complex roots). If all three local extrema were above the x axis, or if there were no local maximum and one minimum above the x axis, there would be no real root (and four complex roots). The same reasoning applies in reverse to polynomial with a negative quartic coefficient.
Graph of a polynomial of degree 4, with 3 critical points and four real roots (crossings of the x axis) (and thus no complex roots). If one or the other of the local minima were above the x axis, or if the local maximum were below it, or if there were no local maximum and one minimum below the x axis, there would only be two real roots (and two complex roots). If all three local extrema were above the x axis, or if there were no local maximum and one minimum above the x axis, there would be no real root (and four complex roots). The same reasoning applies in reverse to polynomial with a negative quartic coefficient.
Quartic function: Solution of x 4 + a x 3 + b x 2 + c x + d = 0 {\displaystyle x^{4}+ax^{3}+bx^{2}+cx+d=0} written out in full. This formula is too unwieldy for general use; hence other methods, or simpler formulas for special cases, are generally used.
Solution of x 4 + a x 3 + b x 2 + c x + d = 0 {\displaystyle x^{4}+ax^{3}+bx^{2}+cx+d=0} written out in full. This formula is too unwieldy for general use; hence other methods, or simpler formulas for special cases, are generally used.

Worked examples

Example 1 — a first encounter with Quartic function

Start with the simplest possible case. Write down what Quartic function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quartic function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quartic function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quartic function

In research
Quartic function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Quartic function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Quartic function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary algebra, Equations, Polynomial functions, so understanding it makes those chapters shorter.
In everyday life
Look for Quartic function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Quartic function in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Quartic function means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Quartic function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quartic function in simple terms?

In algebra, a quartic function is a function of the form f ( x ) = a x 4 + b x 3 + c x 2 + d x + e , {\displaystyle f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,} where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree…

Why does Quartic function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quartic function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quartic function.

Tags

  • Elementary algebra
  • Equations
  • Polynomial functions

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