Question

ПОМОГИТЕ ПОЖАЛУЙСТА!!! Обоснуйте, что данное утверждение неверно: «Уравнение (2 − x) · (x − 5) · (x + 1) · (x − 2) · 2 · x = 0 имеет только следующие корни: −1, 2, 5».

Answer 1

Произведение равно нулю, когда хотя бы один из множителей равен нулю, а остальные имеют смысл.

1) 2-х=0   => x=2
2) x-5=0   => x=5
3) x+1=0  => x=-1
4) 2≠0
5) x=0

таким образом, данное уравнение имеет 4 корня: -1; 0; 2; 5

Answer 2

Invalidity of the Given Statement

The statement "The equation (2 − x) · (x − 5) · (x + 1) · (x − 2) · 2 · x = 0 has only the following roots: -1, 2, 5" is incorrect. Let's delve into the reasons why this statement is invalid.

The given equation is a polynomial equation, and to determine its roots, we need to factorize it and then find the values of x that make the equation equal to zero.

The equation can be factorized as follows: (2 − x) · (x − 5) · (x + 1) · (x − 2) · 2 · x = 0

To find the roots of the equation, we set each factor equal to zero and solve for x: 1. (2 − x) = 0 2. (x − 5) = 0 3. (x + 1) = 0 4. (x − 2) = 0 5. 2 = 0 (This factor will not contribute any real roots as 2 ≠ 0) 6. x = 0 (This factor will contribute a root)

Let's analyze each factor: 1. (2 − x) = 0 gives x = 2 2. (x − 5) = 0 gives x = 5 3. (x + 1) = 0 gives x = -1 4. (x − 2) = 0 gives x = 2 5. 2 = 0 does not give a real root as 2 ≠ 0 6. x = 0 gives x = 0

Therefore, the roots of the given equation are x = 2, 5, -1, and 0. The root x = 2 appears twice due to the factor (x − 2) being repeated.

This analysis demonstrates that the statement claiming the equation has only the roots -1, 2, and 5 is incorrect.