What is the greatest number of zeros the polynomial f(x)=7x

6
−5x
5
+x could have?

1 answer

To find the greatest number of zeros a polynomial can have, we can use the fact that the degree of the polynomial gives us the maximum number of distinct real roots.

The polynomial in question is

\[ f(x) = 7x^6 - 5x^5 + x. \]

The degree of this polynomial is 6, which means that it can have at most 6 roots (counting multiplicities).

To confirm, we can also note that:

  1. A polynomial of degree \( n \) can have at most \( n \) distinct roots.
  2. Roots can be real or complex, but the maximum number of roots (including complex) is still equal to the degree of the polynomial.

Since \( f(x) \) has a degree of 6, the greatest number of zeros (real or complex) that the polynomial can have is:

\[ \text{Greatest number of zeros} = 6. \]

Thus, the answer is that the polynomial \( f(x) \) can have a maximum of 6 zeros.