To determine the total number of roots of the polynomial function [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex], let's follow these steps:
1. Identify the Factored Form:
- The given polynomial is already in its factored form: [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex].
2. Find the Roots:
- The roots of the polynomial are the values of [tex]\( x \)[/tex] that make each factor equal to zero.
- For [tex]\( (x+1) = 0 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[
x = -1
\][/tex]
- For [tex]\( (x-3) = 0 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3
\][/tex]
- For [tex]\( (x-4) = 0 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4
\][/tex]
3. List the Roots:
- The roots of the polynomial are [tex]\( x = -1 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = 4 \)[/tex].
4. Determine the Total Number of Roots:
- Count the roots listed: [tex]\(-1\)[/tex], [tex]\(3\)[/tex], and [tex]\(4\)[/tex].
- There are 3 roots in total.
Therefore, the total number of roots of the polynomial function [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex] is [tex]\( 3 \)[/tex].
