find all complex zeros for the following polynomial function please show all your work
f(x) = x^3 - 125
????????
we have a difference of cubes
x^3 - 125 = (x-5)(x^2 + 5x + 25)
the quadratic has complex roots of
x = (-5 ± √-75)/2
= (-5 ± 5√-3)/2
= (-5 ± 5√3 i )/2
what does the question mark mean????
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To find the complex zeros of the polynomial function f(x) = x^3 - 125, we can use the factoring method. First, let's factor out the difference of cubes formula. The difference of cubes formula is given by:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In our case, a = x and b = 5. Substituting these values into the formula gives:
x^3 - 125 = (x - 5)(x^2 + 5x + 25)
Now, we have factored the polynomial into two parts. To find the zeros, we set each factor equal to zero and solve for x.
First factor: x - 5 = 0
Adding 5 to both sides, we get:
x = 5
Second factor: x^2 + 5x + 25 = 0
This equation cannot be factored any further, so we can solve it using the quadratic formula, which is given by:
x = (-b ± √(b^2 - 4ac))/(2a)
In our case, a = 1, b = 5, and c = 25. Substituting these values into the quadratic formula, we have:
x = (-5 ± √(5^2 - 4(1)(25)))/(2(1))
x = (-5 ± √(-75))/2
The discriminant (b^2 - 4ac) is negative, which means we have complex zeros. Simplifying further, we have:
x = (-5 ± √(-75))/2
x = (-5 ± √(75)i)/2
x = (-5 ± 5√3i)/2
Therefore, the complex zeros of the polynomial function f(x) = x^3 - 125 are:
x = 5
x = (-5 + 5√3i)/2
x = (-5 - 5√3i)/2