- #1
Panphobia
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How would I go about solving an Nth degree polynomial function such that N>=5?
Ax[itex]^{n}[/itex]+Bx[itex]^{n-1}[/itex]+...+Z = 0
Ax[itex]^{n}[/itex]+Bx[itex]^{n-1}[/itex]+...+Z = 0
Panphobia said:That might take a while with this
u(k) = (900-3k)r[itex]^{k-1}[/itex]
s(n) = Σ[itex]_{k=1...n}[/itex]u(k)
find r for which s(5000) = -600,000,000,000
But ok no problem, I guess I will write a program to iterate through all possible values of r.
The Galois group of a general complex polynomial of degree ##n\geq 5## is not solvable. Thus, you won't be able to come up with something like the quadratic formula for ##n\geq 5##. You have to find other ways to find roots.Panphobia said:How would I go about solving an Nth degree polynomial function such that N>=5?
Ax[itex]^{n}[/itex]+Bx[itex]^{n-1}[/itex]+...+Z = 0
Mandelbroth said:The Galois group of a general complex polynomial of degree ##n\geq 5## is not solvable. Thus, you won't be able to come up with something like the quadratic formula for ##n\geq 5##.
You know I meant "in terms of radicals, powers, addition, and subtraction," right? :tongue:jackmell said:Given:
$$ w(z)=a_0+a_1 z+a_2 z^2+\cdots+a_nz^n$$
by Laurent's Expansion Theorem applied to algebraic functions, we can always write down an explicit expression (just like the quadratic formula) for the roots in terms of complex-contour integrals:
$$
w(z)=b\prod_{j=1}^N \left(z-\frac{1}{2k_j\pi i}\mathop{\oint} \frac{z_{j,k}(\zeta)}{\zeta}d\zeta\right)^{k}
$$
However, the integral is in general not solvable in terms of elementary functions so must be evaluated numerically.
Mandelbroth said:I've never thought of that, so I've learned something new today.
An Nth degree polynomial is a mathematical expression that contains one or more terms, with each term having a variable raised to a specific exponent. The degree of the polynomial refers to the highest exponent in the expression.
The degree of a polynomial is determined by the highest exponent in the expression. For example, in the polynomial 2x^3 + 5x^2 + 3x + 1, the degree is 3 because the exponent of the first term is 3.
There are several types of Nth degree polynomials, including linear (degree of 1), quadratic (degree of 2), cubic (degree of 3), and higher degree polynomials such as quartic (degree of 4), quintic (degree of 5), and so on.
To graph an Nth degree polynomial, you can plot points by choosing values for the variable and evaluating the expression. You can also use the leading coefficient and the degree to determine the end behavior and shape of the graph.
Nth degree polynomials are used in many areas of science and engineering, such as physics, chemistry, economics, and computer graphics. They are also used in statistics to model data and make predictions.