Question

X^3 + 2x^2 - x - 2 = 0

Answer 1

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Answer 2

Очевидно х-2 - корень.Поделим на (х+2).                                                                                 
х*х-1=0                                             
х*х=1                                                        
х=-1 или х=-1                                                          
Ответ? х=1, х=-1, х=-2.

Answer 3

To solve the equation $x^3 + 2x^2 - x - 2 = 0$, we can use various methods such as factoring, the rational root theorem, or numerical methods like Newton's method. Let's try to factor the equation first.

Unfortunately, the equation is not easily factorable into simple linear or quadratic factors. So, we'll use the numerical method called the "Rational Root Theorem" to find any rational roots (if they exist), and then we can use synthetic division or long division to factor the polynomial further if needed.

The Rational Root Theorem states that any rational root (if it exists) of a polynomial with integer coefficients $ax^n + bx^{n-1} + \ldots + c$ must be of the form $\frac{p}{q}$, where $p$ is a factor of the constant term $c$ and $q$ is a factor of the leading coefficient $a$.

For our equation $x^3 + 2x^2 - x - 2 = 0$, the leading coefficient is 1, and the constant term is -2.

The factors of -2 are $\pm 1$ and $\pm 2$, and the factors of 1 (leading coefficient) are $\pm 1$.

Now, we can test these possible rational roots one by one to see if any of them satisfy the equation:

1. $x = 1$: $(1)^3 + 2(1)^2 - 1 - 2 = 1 + 2 - 1 - 2 = 0$

So, $x = 1$ is a root of the equation.

Now that we've found one root, $x = 1$, we can factor the polynomial by performing synthetic division or long division.

Using synthetic division: Divide $x^3 + 2x^2 - x - 2$ by $x - 1$ (since $x = 1$ is a root).

luaCopy code
     1 | 1   2   -1   -2
        |    1    3    2
     ------------------
        1   3    2    0

The result is $x^2 + 3x + 2$.

Now, let's solve the quadratic equation $x^2 + 3x + 2 = 0$ by factoring or using the quadratic formula.

$x^2 + 3x + 2 = 0$ factors as $(x + 1)(x + 2) = 0$.

So the remaining roots are $x = -1$ and $x = -2$.

Therefore, the solutions to the original equation $x^3 + 2x^2 - x - 2 = 0$ are $x = 1$, $x = -1$, and $x = -2$.