Question

Решите уравнение y³-2y²+y-2=0 ​

Answer 1

Ответ:

решение смотри на фотографии

Объяснение:

Answer 2

To solve the equation $y^3 - 2y^2 + y - 2 = 0$, we can use different methods like factoring, synthetic division, or numerical methods. In this case, we will use the factoring method.

Step 1: Try to find a factor by substituting some values of $y$. Let's try $y = 1$: $1^3 - 2 \cdot 1^2 + 1 - 2 = 0$ $1 - 2 + 1 - 2 = -2 \neq 0$

Step 2: Since $y = 1$ is not a root, there are no linear factors for this equation.

Step 3: Attempt to use the rational root theorem to find possible rational roots. The rational root theorem states that if $p/q$ is a rational root of the polynomial equation $a_ny^n + a_{n-1}y^{n-1} + \ldots + a_1y + a_0 = 0$, then $p$ is a factor of $a_0$ and $q$ is a factor of $a_n$.

In this case, the polynomial equation is $y^3 - 2y^2 + y - 2 = 0$. The constant term is $a_0 = -2$, and the leading coefficient is $a_n = 1$.

The possible rational roots will be all the divisors/factors of $a_0$ (-2) divided by all the divisors/factors of $a_n$ (1). Possible rational roots: $\pm 1, \pm 2$.

Step 4: Test the possible rational roots using synthetic division or substitution to find the real roots. Let's try $y = 1$: $1^3 - 2 \cdot 1^2 + 1 - 2 = -2 \neq 0$

Let's try $y = -1$: $(-1)^3 - 2 \cdot (-1)^2 - 1 - 2 = 0$

Since $y = -1$ is a root, $y + 1$ is a factor of the polynomial.

Step 5: Perform polynomial division to find the quadratic factor. Using synthetic division:

markdownCopy code
   -1 | 1  -2   1   -2
       |    -1   3   2
       |_________________
         1  -3   4    0

So, $y^3 - 2y^2 + y - 2 = (y + 1)(y^2 - 3y + 4)$.

Step 6: Solve the quadratic factor $y^2 - 3y + 4 = 0$ using the quadratic formula or factoring. The quadratic equation $y^2 - 3y + 4 = 0$ does not have real roots since its discriminant $\Delta = b^2 - 4ac = (-3)^2 - 4(1)(4) = 9 - 16 = -7$ is negative.

Step 7: Conclude the solution. The original equation $y^3 - 2y^2 + y - 2 = 0$ has one real root $y = -1$, and the quadratic factor $y^2 - 3y + 4 = 0$ has two complex roots (non-real). Therefore, the complete solution to the equation is:

$y = -1$ (real root) $y = \frac{3 \pm i\sqrt{7}}{2}$ (complex roots)