Question

X^4+8x^3+17x^2+10x=0

Answer 1

Ответ: x=0 x=-5 x=-2 x=-1

Объяснение:

Answer 2

To solve the equation $x^4 + 8x^3 + 17x^2 + 10x = 0$, we can factor out an $x$ from the equation to simplify it:

$x(x^3 + 8x^2 + 17x + 10) = 0$

Now, we have two possible cases:

Case 1: $x = 0$

If $x = 0$, then the equation is satisfied.

Case 2: $x^3 + 8x^2 + 17x + 10 = 0$

To solve the cubic equation $x^3 + 8x^2 + 17x + 10 = 0$, we can try to find its rational roots using the rational root theorem.

The possible rational roots of the equation are of the form $x = \pm \frac{p}{q}$, where $p$ is a factor of the constant term (10), and $q$ is a factor of the leading coefficient (1).

The factors of 10 are $\pm 1, \pm 2, \pm 5, \pm 10$, and the factors of 1 are $\pm 1$.

Let's try these possible values for $x$ and see if any of them satisfy the equation:

For $x = 1$: $1^3 + 8(1)^2 + 17(1) + 10 = 1 + 8 + 17 + 10 = 36$

For $x = -1$: $(-1)^3 + 8(-1)^2 + 17(-1) + 10 = -1 + 8 - 17 + 10 = 0$

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

Now, we can divide the cubic equation by $x + 1$ using polynomial long division or synthetic division to find the remaining quadratic equation:

$$\begin{array}{c|cccc} x + 1 & x^3 + 8x^2 + 17x + 10 \\ \hline & x^2 + 7x + 10 \\ \end{array}$$

The quotient is $x^2 + 7x + 10$, which is a quadratic equation.

To solve the quadratic equation $x^2 + 7x + 10 = 0$, we can factor it:

$(x + 2)(x + 5) = 0$

This gives us two additional roots: $x = -2$ and $x = -5$.

Therefore, the solutions to the original equation $x^4 + 8x^3 + 17x^2 + 10x = 0$ are $x = 0$, $x = -1$, $x = -2$, and $x = -5$.