Sure, let's find the total number of roots for the polynomial function [tex]\( f(x) = (x+5)^3(x-9)(x+1) \)[/tex] using its factored form.
1. Identify the roots and their multiplicities:
- The factor [tex]\( (x + 5) \)[/tex] gives us a root at [tex]\( x = -5 \)[/tex]. Since this factor is raised to the power of 3, the root [tex]\( x = -5 \)[/tex] has a multiplicity of 3.
- The factor [tex]\( (x - 9) \)[/tex] gives us a root at [tex]\( x = 9 \)[/tex]. This factor is raised to the power of 1, so the root [tex]\( x = 9 \)[/tex] has a multiplicity of 1.
- The factor [tex]\( (x + 1) \)[/tex] gives us a root at [tex]\( x = -1 \)[/tex]. This factor is raised to the power of 1, so the root [tex]\( x = -1 \)[/tex] has a multiplicity of 1.
2. Summarize the roots and their multiplicities:
- Root [tex]\( x = -5 \)[/tex] with multiplicity 3
- Root [tex]\( x = 9 \)[/tex] with multiplicity 1
- Root [tex]\( x = -1 \)[/tex] with multiplicity 1
3. Calculate the total number of roots:
- To find the total number of roots, we add the multiplicities of all the roots:
[tex]\[
3 \, (from \, x = -5) + 1 \, (from \, x = 9) + 1 \, (from \, x = -1) = 5
\][/tex]
So, the total number of roots of the polynomial function [tex]\( f(x) = (x+5)^3(x-9)(x+1) \)[/tex], accounting for their multiplicities, is [tex]\( 5 \)[/tex].
