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What are the real and complex solutions of the polynomial equation?
x4–41x2=–400
To solve this polynomial equation:
x^4 - 41x^2 = -400
Let's make a substitution to simplify the equation. Let y = x^2.
So, the equation becomes:
y^2 - 41y = -400
Now rearrange this equation to get it in standard form:
y^2 - 41y + 400 = 0
Now factor this quadratic equation:
(y - 25)(y - 16) = 0
Now solve for y:
y = 25 or y = 16
Now, substitute back x^2 for y:
x^2 = 25 or x^2 = 16
Take the square root of both sides:
x = ±5 or x = ±4
So the real solutions are x = 5 or x = -5, x = 4 or x = -4.
Therefore, the real solutions of the polynomial equation are x = 5, x = -5, x = 4, x = -4.