Answer :
the given equation has three solutions: x = -10, 1.292, 8.476, 1.046.
To solve the equation 25x³ + 25x² - x - 1 = 0, we are given that -10 is a zero of the function f(x) = 25x³ + 25x² - x - 1.
To find the remaining solutions, we can use polynomial division or synthetic division to divide the polynomial by (x + 10) since -10 is a zero.
Performing the synthetic division:
-10 | 25 25 -1 -1
| -250 0 10 -90
|-----------------------------
0 -225 9 -91
The resulting quotient is 25x² - 225x + 9 with a remainder of -91.
Now we have a quadratic equation: 25x² - 225x + 9 = 0
We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For our quadratic equation, a = 25, b = -225, and c = 9.
Using the quadratic formula, we can calculate the remaining solutions for x:
x = (-(-225) ± √((-225)² - 4(25)(9))) / (2(25))
x = (225 ± √(50625 - 900)) / 50
x = (225 ± √49725) / 50
x ≈ (225 ± 223.17) / 50
So the remaining solutions are:
x ≈ (225 + 223.17) / 50 ≈ 8.476
x ≈ (225 - 223.17) / 50 ≈ 1.046
Therefore, the solutions to the equation 25x³ + 25x² - x - 1 = 0 are approximately:
x = -10, 1.292, 8.476, 1.046.
Final answer:
To solve the equation 25x³ + 25x² - x - 1 = 0, we can use polynomial long division or synthetic division to find the other solutions after given one solution. The solutions include a real number and two complex numbers.
Explanation:
To solve the equation 25x³ + 25x² - x - 1 = 0, we are given that -10 is a zero of the equation. This means that when x = -10, the equation becomes zero. To find the other solutions, we can use polynomial long division or synthetic division to divide the equation by (x - (-10)) or (x + 10). By doing this, we obtain a quadratic equation that we can solve using factoring, the quadratic formula, or completing the square.
The division gives us a quotient of 25x² + 250x + 249. Setting this quotient equal to zero and solving for x, we find that the remaining solutions are not real numbers but complex numbers. So, the solutions to the equation 25x³ + 25x² - x - 1 = 0 are x = -10 and two complex numbers.
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