Let's determine the total number of roots for each polynomial function by examining their factored forms.
1. [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex]
- This polynomial is factored into three linear factors: [tex]\((x+1)\)[/tex], [tex]\((x-3)\)[/tex], and [tex]\((x-4)\)[/tex].
- Each factor contributes one root to the polynomial.
- Therefore, the total number of roots is [tex]\(3\)[/tex].
2. [tex]\( f(x) = (x-6)^2(x+2)^2 \)[/tex]
- This polynomial has two factors raised to a power: [tex]\((x-6)^2\)[/tex] and [tex]\((x+2)^2\)[/tex].
- [tex]\((x-6)^2\)[/tex] means that the root [tex]\(x = 6\)[/tex] has multiplicity 2.
- [tex]\((x+2)^2\)[/tex] means that the root [tex]\(x = -2\)[/tex] has multiplicity 2.
- Therefore, the total number of roots is [tex]\(2 + 2 = 4\)[/tex].
3. [tex]\( f(x) = (x+5)^3(x-9)(x+1) \)[/tex]
- This polynomial has three linear factors and one factor raised to a power: [tex]\((x+5)^3\)[/tex], [tex]\((x-9)\)[/tex], and [tex]\((x+1)\)[/tex].
- [tex]\((x+5)^3\)[/tex] means that the root [tex]\(x = -5\)[/tex] has multiplicity 3.
- [tex]\((x-9)\)[/tex] and [tex]\((x+1)\)[/tex] each contribute one root.
- Therefore, the total number of roots is [tex]\(3 + 1 + 1 = 5\)[/tex].
4. [tex]\( f(x) = (x+2)(x-1)[x-(4+3i)][x-(4-3i)] \)[/tex]
- This polynomial has two real linear factors and two complex conjugate factors: [tex]\((x+2)\)[/tex], [tex]\((x-1)\)[/tex], [tex]\((x-(4+3i))\)[/tex], and [tex]\((x-(4-3i))\)[/tex].
- Each factor contributes one root.
- Therefore, the total number of roots is [tex]\(1 + 1 + 1 + 1 = 4\)[/tex].
To summarize, the total number of roots for each polynomial is:
- [tex]\(f(x) = (x+1)(x-3)(x-4)\)[/tex] has 3 roots.
- [tex]\(f(x) = (x-6)^2(x+2)^2\)[/tex] has 4 roots.
- [tex]\(f(x) = (x+5)^3(x-9)(x+1)\)[/tex] has 5 roots.
- [tex]\(f(x) = (x+2)(x-1)[x-(4+3i)][x-(4-3i)]\)[/tex] has 4 roots.
So, the completed answers for each polynomial are:
- [tex]\(f(x) = (x+1)(x-3)(x-4)\)[/tex] [tex]\(\boxed{3}\)[/tex]
- [tex]\(f(x) = (x-6)^2(x+2)^2\)[/tex] [tex]\(\boxed{4}\)[/tex]
- [tex]\(f(x) = (x+5)^3(x-9)(x+1)\)[/tex] [tex]\(\boxed{5}\)[/tex]
- [tex]\(f(x) = (x+2)(x-1)[x-(4+3i)][x-(4-3i)]\)[/tex] [tex]\(\boxed{4}\)[/tex]
