What is the greatest number of 0s this polynomial can have

Question

f(x)=7x^6-5x^5+x

Answer 1

To determine the greatest number of zeros that the polynomial $f(x)=7x^6-5x^5+x$ can have, we can start by factoring the polynomial.

First, we can factor out the common term $x$:

$$f(x)=x(7x^5-5x^4+1)$$

This shows that $x=0$ is one root, contributing one zero. Now we need to analyze the polynomial $7x^5-5x^4+1$ to find the number of additional zeros it may have.

A polynomial of degree $n$ can have at most $n$ zeros. The polynomial $7x^5-5x^4+1$ is a 5th degree polynomial, which means it can have up to 5 real roots (including complex roots).

Thus, taking into account the root $x=0$ from the factored form, the total greatest number of zeros of $f(x)$ is:

$$1\text{ (for }x=0\text{)}+5\text{ (from }7x^5-5x^4+1\text{)}=6$$

Therefore, the greatest number of zeros that the polynomial $f(x)=7x^6-5x^5+x$ can have is $\boxed{{\displaystyle 6}}$.