To determine the total number of roots of a polynomial function given in its factored form, you need to identify the roots and their respective multiplicities. The multiplicity of a root indicates how many times that particular root appears in the factorization of the polynomial. To get the total number of roots, simply sum the multiplicities of all the distinct roots.
Let's start with the polynomial function given:
[tex]\[ f(x) = (x+5)^3(x-9)(x+1) \][/tex]
### Step-by-Step Solution:
1. Identify the Roots and Their Multiplicities:
- The factor [tex]\( (x+5)^3 \)[/tex] gives the root [tex]\( x = -5 \)[/tex] with multiplicity 3.
- The factor [tex]\( (x-9) \)[/tex] gives the root [tex]\( x = 9 \)[/tex] with multiplicity 1.
- The factor [tex]\( (x+1) \)[/tex] gives the root [tex]\( x = -1 \)[/tex] with multiplicity 1.
2. Sum the Multiplicities:
- The root [tex]\( x = -5 \)[/tex] has multiplicity 3.
- The root [tex]\( x = 9 \)[/tex] has multiplicity 1.
- The root [tex]\( x = -1 \)[/tex] has multiplicity 1.
[tex]\[
\text{Total number of roots} = 3 (from \, x=-5) + 1 (from \, x=9) + 1 (from \, x=-1)
\][/tex]
3. Calculate the Total:
[tex]\[
3 + 1 + 1 = 5
\][/tex]
Therefore, the total number of roots of the polynomial function [tex]\( f(x) = (x+5)^3(x-9)(x+1) \)[/tex], taking into account the multiplicity of each root, is [tex]\( 5 \)[/tex].
