Question

Расскажите пожалуйста теорему виетта, и покажите на примере, а то не очень понимаю

Answer 1

Решение приложено к снимку

Answer 2

The Vieta's Theorem

Vieta's theorem is a fundamental result in algebra that relates the coefficients of a polynomial to the sums and products of its roots. It provides a way to find relationships between the roots and coefficients of a polynomial equation.

Let's consider a polynomial equation of degree n:

a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ = 0

where a_n, a_n-1, ..., a₁, a₀ are the coefficients of the polynomial, and x is the variable.

Vieta's theorem states that for this polynomial equation, the sum of the roots is equal to the negation of the coefficient of the second-to-last term divided by the coefficient of the leading term, and the product of the roots is equal to the constant term divided by the coefficient of the leading term.

Let's illustrate this with an example:

Consider the quadratic equation:

2x² - 5x + 3 = 0

According to Vieta's theorem, the sum of the roots is equal to the negation of the coefficient of the second-to-last term divided by the coefficient of the leading term. In this case, the sum of the roots is:

x₁ + x₂ = -(-5) / 2 = 5/2

The product of the roots is equal to the constant term divided by the coefficient of the leading term. In this case, the product of the roots is:

x₁ * x₂ = 3 / 2 = 3/2

So, for the quadratic equation 2x² - 5x + 3 = 0, the sum of the roots is 5/2 and the product of the roots is 3/2.

This theorem can be extended to equations of higher degrees as well, where the relationships between the roots and coefficients become more complex. Vieta's theorem provides a powerful tool for solving polynomial equations and understanding the properties of their roots.