How to find the roots of a quartic polynomial?

To find the roots of a quartic polynomial, we can use the quartic formula or factorise it.

The quartic formula is a lengthy and complex formula that can be used to find the roots of any quartic polynomial. It is given by:

x = [-b ± √(b² - 4ac - 2ad²) ] / 2a or x = [-b ± i√(4ad² - b² + 2ac)] / 2a

where a, b, c, and d are the coefficients of the quartic polynomial ax⁴ + bx³ + cx² + dx + e. This formula can be quite difficult to use, especially when the coefficients are large or complex. Understanding the basics of rational functions might provide additional insights into managing such complex calculations.

Alternatively, we can factorise the quartic polynomial into two quadratic factors and then solve for the roots using the quadratic formula. This method is often easier and quicker than using the quartic formula. To factorise a quartic polynomial, we can use a variety of techniques such as grouping, substitution, or trial and error. Learning about polyhedra can offer further understanding of factorisation techniques used in higher-dimensional equations.

For example, consider the quartic polynomial x⁴ - 5x² + 4. We can factorise this as (x² - 4)(x² - 1) = 0. Then, using the quadratic formula, we can solve for the roots of each quadratic factor:

x² - 4 = 0 => x = ±2
x² - 1 = 0 => x = ±1

Therefore, the roots of the quartic polynomial are x = ±2 and x = ±1. For those interested in the underlying calculus involved in solving these equations, exploring the introduction to derivatives could be very beneficial.

A-Level Maths Tutor Summary: To find a quartic polynomial's roots, you can use the complex quartic formula or factorise it into simpler quadratic equations. The quartic formula involves coefficients a, b, c, and d and requires careful calculation. Factorisation splits the polynomial into easier parts, allowing for simpler solutions. For example, breaking down x⁴ - 5x² + 4 leads to roots x = ±2 and x = ±1.