The function f(x) = x^3 + 3x^2 + 3x + 9 has 3 complex zeros, which includes both real and imaginary zeros. There are no obvious rational zeros based on the Rational Root Theorem, and the exact zeros cannot be determined without numerical methods or graphing. All zeros are potentially complex or imaginary since there are no rational zeros.
The function f(x) = x^3 + 3x^2 + 3x + 9 is a cubic polynomial, and by the Fundamental Theorem of Algebra, it will have 3 complex zeros (which includes real zeros and imaginary zeros).
The possible real zeros of a polynomial function can be found using the Rational Root Theorem.
However, for this particular function, there are no rational zeros, as no integer factors of 9 (the constant term) will satisfy the polynomial equation.
To find the exact zeros of this function, one would typically use numerical methods or graphing technology, as the cubic does not factor nicely.
For this function, since it has odd-degree and positive leading coefficient, it will have at least one real zero.
No real zeros can be observed through the Rational Root Theorem, and therefore, all zeros are potentially complex or imaginary.
The possible imaginary zeros are typically calculated as complex conjugate pairs, when you find one non-real zero its conjugate is also a zero of the function.
Since we have no easy method of finding the exact zeros in this case, we cannot list the possible imaginary zeros without further calculations or graphing.
The probable question may be:
10. Use the same function to answer #16-18.
16) How many complex zeros, real zeros, and imaginary zeros does the function have? (3 points) f(x) = x3 + 3x2 + 3x + 9
Complex Zeros (total zeros): ________
Possible Real Zeros: _________
Possible Imaginary Zeros: ________
17) What are the possible rational zeros of the function? (± list) (3 points)
18) Find all the zeros. (4 points)