What are the real and complex solutions of the polynomial equation? x^4-41x^2=-400
To find the real and complex solutions of the polynomial equation x^4 - 41x^2 = -400, we can substitute a variable for x^2. Let's say u = x^2. Then the equation becomes:
u^2 - 41u = -400.
Now we have a quadratic equation in terms of u:
u^2 - 41u + 400 = 0.
To solve this quadratic equation, we can factor it as (u - 25)(u - 16) = 0.
Setting each factor to zero, we get u = 25 and u = 16. Since u = x^2, we get x^2 = 25 and x^2 = 16.
Taking the square root of both sides, we find the real solutions for x:
x = ±5 and x = ±4.
For the complex solutions, we use the fact that every non-real complex number has a pair of conjugate solutions. Therefore, the pair of complex solutions will be ±5i and ±4i.
In summary, the real solutions are x = ±5 and x = ±4, while the complex solutions are x = ±5i and x = ±4i.