Finding Roots of A Quintic Equation
Finding the roots of a given polynomial has been a prominent mathematical problem.
Solving linear, quadratic, cubic and quartic equations by factorization into radicals is fairly straightforward, no matter whether the roots are rational or irrational, real or complex; there are also formulae that yield the required solutions. However, there is no formula for general quintic equations over the rationals in terms of radicals; this is known as the Abel–Ruffini theorem, first published in 1824, which was one of the first applications of group theory in algebra. This result also holds for equations of higher degrees. An example quintic whose roots cannot be expressed by radicals is This quintic is in Bring–Jerrard normal form.
As a practical matter, exact analytic solutions for polynomial equations are often unnecessary, and so numerical methods such as Laguerre's method or the Jenkins-Traub method are probably the best way of obtaining solutions to general quintics and higher degree polynomial equations that arise in practice. However, analytic solutions are sometimes useful for certain applications, and many mathematicians have tried to develop them.
Read more about this topic: Quintic Function
Famous quotes containing the words finding, roots and/or equation:
“There comes a point in many people’s lives when they can no longer play the role they have chosen for themselves. When that happens, we are like actors finding that someone has changed the play.”
—Brian Moore (b. 1921)
“Though of erect nature, man is far above the plants. For man’s superior part, his head, is turned toward the superior part of the world, and his inferior part is turned toward the inferior world; and therefore he is perfectly disposed as to the general situation of his body. Plants have the superior part turned towards the lower world, since their roots correspond to the mouth, and their inferior parts towards the upper world.”
—Thomas Aquinas (c. 1225–1274)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)