Final answer:
The function f(x) = 7x⁵-3x³+5x+2 has 1 real root and 4 imaginary roots.
Step-by-step explanation:
A polynomial function is said to have real and imaginary roots. Real roots are the values of x that make the function equal to zero. Imaginary roots are values that cannot be expressed as real numbers. To find the number and types of roots of a polynomial function, we need to use the Fundamental Theorem of Algebra.
For the given function f(x) = 7x⁵-3x³+5x+2, which is a polynomial of degree 5, we can determine the number of real roots by counting the number of sign changes in the coefficients of the function. In this case, there are two sign changes, indicating that there are 2 or 0 real roots.
Since the degree is odd and there are two sign changes, f(x) has exactly 1 real root. Therefore, the function f(x) = 7x⁵-3x³+5x+2 has 1 real root and 4 imaginary roots.