Answer :
Final answer:
The function f(x) = 7x^3 - x^2 + 10x - 4 can have either no imaginary numbers or two imaginary numbers, as complex roots (if any) always occur in conjugate pairs for polynomials with real coefficients.
Explanation:
The student asked about the possible number of imaginary numbers for the function f(x) = 7x3 - x2 + 10x - 4. This function is a cubic polynomial, and the Fundamental Theorem of Algebra tells us that a cubic polynomial will have 3 roots in total, which could be real or complex numbers. According to the nature of complex roots, if they occur, they must occur in conjugate pairs.
Therefore, for the given cubic polynomial, we can have the following scenarios regarding the number of imaginary numbers (complex roots):
- No imaginary numbers (all roots are real)
- Two imaginary numbers (one real root and a pair of complex conjugate roots)
It's not possible to have just one or three imaginary numbers because complex roots always come in pairs due to the real coefficients of the polynomial.
