To determine the total number of roots of each polynomial function using the factored form, let’s analyze each polynomial function given:
1. For the polynomial function [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex]:
- We have three factors: [tex]\( (x+1) \)[/tex], [tex]\( (x-3) \)[/tex], and [tex]\( (x-4) \)[/tex].
- Each factor can set to zero to find the roots:
- [tex]\( x+1 = 0 \)[/tex] gives the root [tex]\( x = -1 \)[/tex].
- [tex]\( x-3 = 0 \)[/tex] gives the root [tex]\( x = 3 \)[/tex].
- [tex]\( x-4 = 0 \)[/tex] gives the root [tex]\( x = 4 \)[/tex].
- Therefore, the polynomial [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex] has 3 distinct roots.
2. For the polynomial function [tex]\( f(x) = (x-6)^2 (x+2)^2 \)[/tex]:
- We have two distinct factors, each raised to a power of 2:
- [tex]\( (x-6)^2 \)[/tex]: This factor indicates that [tex]\( x = 6 \)[/tex] is a root with multiplicity 2.
- [tex]\( (x+2)^2 \)[/tex]: This factor indicates that [tex]\( x = -2 \)[/tex] is a root with multiplicity 2.
- The total number of roots is counted by summing the multiplicities of the distinct roots:
- [tex]\( x = 6 \)[/tex] contributes 2 roots.
- [tex]\( x = -2 \)[/tex] contributes 2 roots.
- Therefore, the polynomial [tex]\( f(x) = (x-6)^2 (x+2)^2 \)[/tex] has a total of 4 roots when multiplicities are counted.
Summarizing:
- The polynomial [tex]\( f(x) = (x+1)(x-3)(x-4) \)[/tex] has 3 roots.
- The polynomial [tex]\( f(x) = (x-6)^2 (x+2)^2 \)[/tex] has 4 roots.
