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zorro
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Homework Statement
How to solve x^{4} + x^{3} + x^{2} + x + 1= 0 ?
Divide by LaTeX Code: x^2 , yielding
LaTeX Code: x^2 + x + 1 + x^{-1} + x^{-2} = 0 .
Now let LaTeX Code: y = x + x^{-1} .
With this substitution, the equation becomes
LaTeX Code: y^2 + y - 1 = 0 .
Solve for y (two roots), then solve for x (four roots).
A 4th degree equation is an algebraic equation in which the highest exponent of the variable is 4. It can be written in the form ax^{4} + bx^{3} + cx^{2} + dx + e = 0, where a, b, c, d, and e are constants and x is the variable.
To solve a 4th degree equation, you can use various methods such as factoring, substitution, or the quadratic formula. However, in most cases, it is solved using numerical methods or computer algorithms.
Yes, every 4th degree equation can be solved. However, the solutions may not always be real numbers. Some equations may have complex solutions, while others may have irrational or imaginary solutions.
A 4th degree equation can have up to four solutions, as the highest exponent is 4. However, some equations may have fewer solutions or no real solutions at all, depending on the coefficients and the nature of the equation.
4th degree equations are important in science because they can be used to model and solve various real-life problems, especially in fields such as physics, engineering, and economics. They can also help in understanding complex systems and their behavior.