More specifically, for a polynomial equation of degree n, if a complex root R appears k times (k ≥ 2), then we call R the k-fold root of the equation. This means that in the factorization of polynomial equations, polynomials can be written in the form of (x-r) k, where x-r is a linear factor and k represents its multiplicity.
For example, consider the polynomial equation f (x) = (x-1) (x-1) (x+2) (x-3) 3. In this equation, the root 1 appears twice, and the root -2 and root -3 appear once and three times respectively. Therefore, the root 1 is double, the root -2 is single and the root 3 is three.
Multiple roots play an important role in algebra, calculus and equation solving. The concept of multiple roots is often involved in factorization of polynomials, tangent and extreme value of curves.
Calculation method of multiple roots in mathematics
Calculating the roots of polynomial equations and their multiples usually requires algebraic knowledge and related numerical calculation methods. The following are some common methods for calculating duplicate roots:
1. Algebraic method: For a polynomial of high degree, the multiplicity of its roots can be calculated by algebraic method. First, find the roots of polynomials and factorize them. Then, by observing the power of each factor in factorization, the multiplicity of the root is determined. For example, a root appears twice in factorization, which means it is a multiple root.
2. Derivative method: The concept of derivative in calculus can help to calculate the multiple roots of polynomial functions. For polynomial function f(x), calculate its derivative f'(x). If a root R is the common root of f(x) and f'(x), then it is a multiple root. Continue to calculate the roots of f'(x) and repeat this process until all the multiple roots are found.
3. Numerical method: If you only care about the numerical solution of polynomial equation and don't need accurate symbolic representation, you can use numerical calculation method to estimate roots and root multiplicity. For example, using numerical calculation software or algorithms (such as Newton iteration method, dichotomy, etc. ) can approximate the root of a polynomial and judge the multiplicity by observing the neighborhood of the root.
Calculating multiple roots may require the support of some mathematical background and calculation tools. For complex polynomial equations, it may be very difficult or even impossible to calculate multiple roots. In practical application, computer algebra system or numerical calculation software is usually used to solve the roots and multiple roots of polynomial equations.
When it comes to computing multiple roots, we can consider the following examples:
Example: Calculate the roots and multiplicity of polynomial equation F (x) = x4-6x3+13x2-12x+4.
Solution:
In order to calculate the roots and multiples of polynomial equations, we can use factorization and derivation.
1. Factorization:
First of all, you can try to factorize the polynomial and see the multiplicity of the root.
f(x)=(x- 1)(x- 1)(x-2+I)(x-2-I)
According to factorization, the roots of polynomials include two multiple roots r = 1 and two roots r = 2 I, which are compound yokes. Where the root r = 1 is a double root and the root r = 2 I is a single root.
2. Derivation method:
We can also calculate the derivative f'(x) of polynomial f(x), and then find the common root to determine the multiple root.
f '(x)= 4x^3- 18x^2+26x- 12
Calculating the root of f'(x), we find that there is only one common root x = 1. This means that the root x = 1 is the common root of equations f(x) and f'(x), so its multiplicity is at least twice.
To sum up, the roots and multiplicity of polynomial equation f (x) = x 4-6x 3+13x 2-12x+4 are as follows:
-the root r = 1 is a multiple root.
-The root r = 2 is a single root.
-The root r = 2 I is two single-line roots.
This example shows how to calculate the roots and multiples of polynomial equations by factorization and derivative methods.