To determine the total number of roots of the polynomial function given in the factored form
[tex]\[ f(x) = (x + 1)(x - 3)(x - 4), \][/tex]
we follow these steps:
1. Identify the factors of the polynomial:
The polynomial is already given in its factored form: [tex]\((x + 1)\)[/tex], [tex]\((x - 3)\)[/tex], and [tex]\((x - 4)\)[/tex].
2. Determine the roots from the factors:
Each factor of the polynomial can be set to zero to find the roots:
[tex]\[
\begin{array}{l}
x + 1 = 0 \implies x = -1 \\
x - 3 = 0 \implies x = 3 \\
x - 4 = 0 \implies x = 4 \\
\end{array}
\][/tex]
3. List the roots:
The roots of the polynomial are:
[tex]\[
x = -1, x = 3, \text{and } x = 4.
\][/tex]
4. Count the number of roots:
Since the polynomial has three distinct factors, we have three distinct roots:
[tex]\[
-1, 3, \text{and } 4.
\][/tex]
Therefore, the total number of roots for the polynomial function [tex]\( f(x) = (x + 1)(x - 3)(x - 4) \)[/tex] is [tex]\(3\)[/tex].
