Given a polynomial function of degree 5, \(f(x) = 2x^5 - 3x^4 + 2x^3 - x^2 + 3x - 2\), find the maximum number of real roots that this function can have.
Answer
Hence, the maximum number of real roots that the function can have is 5, which would occur if all of the roots are real and none are complex.
Step 1 ：By the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n roots. These roots can either be real or complex.
Step 2 ：Since the given function is a polynomial of degree 5, it will have exactly 5 roots. However, complex roots always occur in conjugate pairs. Therefore, if there are any complex roots, there will be an even number of them.
Step 3 ：Hence, the maximum number of real roots that the function can have is 5, which would occur if all of the roots are real and none are complex.
