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What are the real and complex solutions of the polynomial equation?
x4–41x2=–400
To solve the equation x^4 - 41x^2 = -400, we can let y = x^2.
Substituting y into the equation, we have y^2 - 41y + 400 = 0.
We can then factor the quadratic equation:
(y - 25)(y - 16) = 0.
Now we can set each factor equal to zero and solve for y:
y - 25 = 0 --> y = 25
y - 16 = 0 --> y = 16
Since y = x^2, we substitute the values of y back into the equation and solve for x:
x^2 = 25 --> x = ±√25 --> x = ±5.
x^2 = 16 --> x = ±√16 --> x = ±4.
Therefore, the real solutions of the polynomial equation x^4 - 41x^2 = -400 are x = -5, -4, 4, and 5.