Final answer:
To find the complex zeros of the polynomial function f(x) = x⁴ + 122x² + 121 and write it in factored form, we can solve the quadratic equation y² + 122y + 121 = 0, where y is substituted for x². Using the quadratic formula, we find the possible values for y as -1.17 and -121.17. Substituting y back into x² = y gives us the complex zeros as ±√(-1.17).
Step-by-step explanation:
To find the complex zeros of the polynomial function f(x) = x⁴ + 122x² + 121 and write it in factored form, we can first set the function equal to zero:
x⁴ + 122x² + 121 = 0
This is a quadratic equation in terms of x². We can substitute x² with a variable, say, y, and solve for y:
y² + 122y + 121 = 0
Now, we can use the quadratic formula to find the values of y:
y = (-122 ± √(122² - 4(1)(121))) / (2(1))
Simplifying, we get:
y = (-122 ± √(14944 - 484)) / 2
y = (-122 ± √14460) / 2
y = (-122 ± 120.17) / 2
So the possible values for y are -1.17 and -121.17.
Now, we can substitute y back into the equation x² = y, and solve for x:
x² = -1.17
x = ±√(-1.17)
This means that the quadratic equation has no real solutions. Therefore, the complex zeros of the polynomial function f(x) = x⁴ + 122x² + 121 are ±√(-1.17).