Question

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

Answer 1

x⁴ - 10x³ + 250x - 625 = 0

x⁴ - 625 + 10x³ + 250x = 0

(x² - 25)(x² + 25) + 10x(x² - 25) = 0

(x² - 25)(x² + 10x + 25) = 0

(x+5)(x-5)(x-5) = 0

x1 = -5

x2 = 5

Ответ: -5; 5.

Answer 2

The equation you've given is a quartic equation, which means it's a polynomial equation of degree 4. To solve this equation, you can use various methods, such as factoring, the Rational Root Theorem, or numerical methods.

In this case, I'll demonstrate how to solve it using the Rational Root Theorem and synthetic division. The Rational Root Theorem states that if a rational number p/q is a root of the polynomial equation, where p and q are coprime integers, then p must be a factor of the constant term (in this case, 625), and q must be a factor of the leading coefficient (in this case, 1).

The factors of 625 are ±1, ±5, ±25, ±125, and ±625, and the factors of 1 are ±1.

Let's test these potential rational roots using synthetic division:

1. Test x = 1: Synthetic Division:

   Copy code
   1 | 1  -10  250  -625
      |    1   -9   241
   

   The remainder is 241, so x = 1 is not a root.

2. Test x = -1: Synthetic Division:

   diffCopy code
   -1 | 1  -10  250  -625
       |   -1   11  -261
   

   The remainder is -261, so x = -1 is not a root.

3. Test x = 5: Synthetic Division:

   Copy code
   5 | 1  -10  250  -625
      |    5   -25   125
   

   The remainder is 125, so x = 5 is not a root.

4. Test x = -5: Synthetic Division:

   diffCopy code
   -5 | 1  -10  250  -625
        |   -5    75  -625
   

   The remainder is -625, so x = -5 is not a root.

5. Test x = 25: Synthetic Division:

   markdownCopy code
   25 | 1  -10  250  -625
       |  25   375  15625
   

   The remainder is 15625, so x = 25 is not a root.

6. Test x = -25: Synthetic Division:

   diffCopy code
   -25 | 1  -10  250  -625
        |  -25   625 -16875
   

   The remainder is -16875, so x = -25 is not a root.

7. Test x = 125: Synthetic Division:

   markdownCopy code
   125 | 1  -10  250  -625
        |  125  18750  2187500
   

   The remainder is 2187500, so x = 125 is not a root.

8. Test x = -125: Synthetic Division:

   diffCopy code
   -125 | 1  -10  250  -625
         |   -125   15625  -1953125
   

   The remainder is -1953125, so x = -125 is not a root.

None of the potential rational roots tested resulted in a remainder of 0, which means this quartic equation does not have any rational roots. Therefore, to find the solutions, you would need to use numerical methods or a calculator. These methods would approximate the solutions, which would likely involve using numerical techniques like Newton's method or a graphing calculator to find the roots.