Roots of 4 Degree Polynomial Calculator

What is a 4th Degree Polynomial?

A 4th degree polynomial is a mathematical expression that consists of a sum of terms, each of which is a coefficient multiplied by a variable raised to a non-negative integer power. The highest power of the variable in the polynomial is 4, which makes it a quartic equation.

The general form of a 4th degree polynomial is:

Where:

• a, b, c, d, and e are coefficients
• x is the variable
• a ≠ 0 (since it's a 4th degree polynomial)

The roots of the polynomial are the values of x that satisfy the equation. A quartic equation can have up to four real roots or two pairs of complex conjugate roots.

How to Find the Roots of a 4th Degree Polynomial

Finding the roots of a quartic equation is more complex than solving quadratic or cubic equations. There are several methods to find the roots:

1. Factorization: Attempt to factor the polynomial into simpler polynomials whose roots can be found easily.
2. Substitution: Use substitution to simplify the equation into a form that can be solved.
3. Numerical Methods: Use iterative numerical methods to approximate the roots.
4. Ferrari's Solution: A method developed by Ludovico Ferrari in the 16th century that reduces the quartic to a cubic resolvent.

This calculator uses a combination of numerical methods to find the roots of the quartic equation.

The Formula for Finding Roots

The general solution for a quartic equation is complex and involves solving a cubic resolvent. The calculator uses an iterative numerical approach to find the roots.

The formula used by the calculator is based on the following steps:

1. Depress the quartic equation to eliminate the x³ term.
2. Solve the resulting cubic equation to find a parameter.
3. Use the parameter to find the roots of the quartic equation.

The exact formula is complex and involves multiple steps, but the calculator handles all the calculations for you.

Worked Example

Let's find the roots of the quartic equation: x⁴ - 5x² + 4 = 0.

Using the calculator, we can input the coefficients:

• a = 1
• b = 0
• c = -5
• d = 0
• e = 4

The calculator will find the roots:

• x = 1
• x = -1
• x = 2
• x = -2

These are the four real roots of the equation.

Limitations of the Calculator

The calculator uses numerical methods to find the roots of the quartic equation. While it provides accurate results for most cases, there are some limitations:

• Complex roots are shown in the form of real and imaginary parts.
• The calculator may not find all roots if the equation has multiple roots of the same value.
• For very large coefficients, the calculator may not provide precise results due to floating-point arithmetic limitations.