What are the real and complex solutions of the polynomial equation?x4–41x2=–400
To find the solutions of the polynomial equation x^4 - 41x^2 = -400, we can rewrite it as x^4 - 41x^2 + 400 = 0.
Now, let's let y = x^2. Substituting y into the equation, we get y^2 - 41y + 400 = 0.
This is a quadratic equation, and we can solve it by factoring or using the quadratic formula. In this case, it factors nicely as (y - 16)(y - 25) = 0.
So, y - 16 = 0 or y - 25 = 0.
If y - 16 = 0, then y = 16. Substituting back x^2 for y, we have x^2 = 16.
Taking the square root of both sides, we get x = ±4.
If y - 25 = 0, then y = 25. Substituting back x^2 for y, we have x^2 = 25.
Taking the square root of both sides, we get x = ±5.
Therefore, the real solutions of the polynomial equation are x = ±4, ±5.
As for the complex solutions, there are none in this case since all the solutions are real.