To determine the total number of roots of the polynomial function [tex]$f(x)=(x+1)(x-3)(x-4)$[/tex], we can take the following steps:
1. Identify the factors of the polynomial: The given polynomial [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex] is already in factored form. Each factor corresponds to a potential root of the polynomial.
2. Find the roots by setting each factor equal to zero:
[tex]\[
\begin{aligned}
& \text{For the factor } (x+1): \\
& x+1 = 0 \\
& x = -1 \\
\end{aligned}
\][/tex]
[tex]\[
\begin{aligned}
& \text{For the factor } (x-3): \\
& x-3 = 0 \\
& x = 3 \\
\end{aligned}
\][/tex]
[tex]\[
\begin{aligned}
& \text{For the factor } (x-4): \\
& x-4 = 0 \\
& x = 4 \\
\end{aligned}
\][/tex]
3. List all the roots: The polynomial [tex]\( f(x) \)[/tex] has roots at [tex]\( x = -1, x = 3, \)[/tex] and [tex]\( x = 4 \)[/tex].
4. Count the distinct roots: The total number of distinct roots is simply the count of the different values obtained. Here, we have the roots [tex]\( -1, 3, \)[/tex] and [tex]\( 4 \)[/tex], which are all distinct.
Therefore, the total number of distinct roots of the polynomial function [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex] is:
[tex]\[
3
\][/tex]
Summarizing, the roots of the polynomial are [tex]\( -1, 3, \)[/tex] and [tex]\( 4 \)[/tex], and there are 3 distinct roots in total.
