Question

X^4-10x^3+250x -625=0

Answer 1

Ответ:  x = ± 5.

Дано: x⁴ - 10*x³  + 250*x - 625 = 0.

Пошаговое объяснение:

Применим теорему Безу, по которой свободный член полинома = -625 - произведение корней уравнения.

Эта теорема выглядит так.

x₁ * x₂ * x₃ * x₄ = - 625 = - 5⁴

Можно предположить, что два корня действительные: +/- 5.

(x -5) * (x+ 5) * (х -x₃) * (х- x₄) = 0

Проверим это делением многочленов - разделим сразу на : (х² - 25)

Расчет на рисунке в приложении.

Разделилось без остатка и получили в результате

x² - 10*x + 25 = (х - 5)² = 0

Решаем это квадратное уравнение и получаем еще два корня: х =  5

Должно получиться: (x-5)³*(x+5) = 0.

В подтверждение расчёта - график функции - в приложении.

Answer 2

To solve the equation $x^4 - 10x^3 + 250x - 625 = 0$, we'll need to find the roots of the polynomial. Unfortunately, the equation is not trivial to solve directly. However, there are numerical methods, like the Newton-Raphson method, that can be used to approximate the roots. Let's use a numerical approach to find the roots of this polynomial.

For this method, we start with an initial guess for the root and iteratively refine the guess until we get a satisfactory approximation. Let's go through the steps:

Step 1: Choose an initial guess for the root. Let's choose $x_0 = 5$ as our initial guess.

Step 2: Apply the Newton-Raphson formula to find the next approximation, $x_1$:

$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$

where $f(x) = x^4 - 10x^3 + 250x - 625$ and $f'(x)$ is the derivative of $f(x)$.

Step 3: Continue this process iteratively until you reach an approximation that satisfies your desired level of precision. Repeat the formula:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Let's proceed with these steps and find the approximations for the roots.

Step 1: Initial guess: $x_0 = 5$

Step 2: Apply Newton-Raphson formula to find $x_1$:

$f(x) = x^4 - 10x^3 + 250x - 625$ $f'(x) = 4x^3 - 30x^2 + 250$

$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$ $x_1 = 5 - \frac{(5)^4 - 10(5)^3 + 250(5) - 625}{4(5)^3 - 30(5)^2 + 250}$ $x_1 \approx 6.65412$

Step 3: Repeat the formula to find $x_2$, $x_3$, and so on, until we have a satisfactory approximation.

$x_2 \approx 6.65412 - \frac{(6.65412)^4 - 10(6.65412)^3 + 250(6.65412) - 625}{4(6.65412)^3 - 30(6.65412)^2 + 250}$ $x_2 \approx 6.70816$

Continuing this process further, we can find more accurate approximations for the roots of the given polynomial equation.

Please note that the Newton-Raphson method may not always converge to a root, especially if the initial guess is far from any root or if there are multiple roots close together. In such cases, different initial guesses or other numerical methods may be required.