Sum of roots:
The sum of all roots is equal to the negative number b/a, that is, [x _1+x _ 2+x _ 3+\ ldots+x _ n =-\ frac {b} {a}].
Product of roots:
The product of all roots is equal to k/a, that is, [x _1\ cdot x _ 2 \ cdot x _ 3 \ cdot \ ldots \ cdot x _ n = \ frac {k} {a}].
Double root products:
The sum of the products of all pairs of different roots can be expressed by the combination of coefficients. For any i≠j, the sum of [x_i\cdot x_j] is equal to c/a, [x _ i \ cdot x _ j+x _ i \ cdot x _ k+\ ldots = \ frac {c} {a}], where I, j, and.
The Function of Vieta Theorem:
1, solving the root of polynomial equation:
Vieta theorem can be used to find the roots of polynomial equations, including univariate polynomial equations and multivariate polynomial equations. By calculating the sum, product and cross product of roots, we can get important information about roots.
2. Construct a polynomial equation:
In some cases, the roots and partial coefficients of polynomials are known, and Vieta theorem can be used to construct polynomial equations with these roots. This is usually used in engineering, physics and mathematical modeling.
3, decomposition polynomial:
Vieta theorem can help us to decompose polynomials into simpler factors, especially when we know the roots of polynomials, we can use it to decompose polynomials into products of linear factors.
4. Study the properties of polynomials:
Vieta theorem provides an important relationship between the roots and coefficients of polynomials, which is very useful for studying the properties and behaviors of polynomials. For example, the symmetry and stability of polynomials can be inferred by the sum, product and cross product of roots.