To determine the total number of roots of the polynomial function [tex]\( f(x) = (x - 6)^2 (x + 2)^2 \)[/tex] using its factored form, follow these steps:
1. Identify the roots from the factored form:
- Notice that the polynomial is provided in factored form as [tex]\( (x - 6)^2 (x + 2)^2 \)[/tex].
- The factors indicate the roots of the polynomial.
2. Understand the exponents of each factor:
- The factor [tex]\( (x - 6)^2 \)[/tex] suggests that [tex]\( x = 6 \)[/tex] is a root. The exponent 2 indicates that [tex]\( x = 6 \)[/tex] is a double root, meaning it occurs with a multiplicity of 2.
- Similarly, the factor [tex]\( (x + 2)^2 \)[/tex] indicates that [tex]\( x = -2 \)[/tex] is a root with a multiplicity of 2.
3. List all roots with their multiplicities:
- For [tex]\( x = 6 \)[/tex], the multiplicity is 2.
- For [tex]\( x = -2 \)[/tex], the multiplicity is 2.
4. Calculate the total number of roots including their multiplicities:
- The multiplicity of a root is the number of times it counts as a root.
- Therefore, [tex]\( x = 6 \)[/tex] counts as 2 roots and [tex]\( x = -2 \)[/tex] also counts as 2 roots.
5. Sum the multiplicities:
- The total number of roots is the sum of the multiplicities:
[tex]\[
2 \text{ (for } x = 6 \text{)} + 2 \text{ (for } x = -2 \text{)} = 4
\][/tex]
Thus, the total number of roots of the polynomial function [tex]\( f(x) = (x - 6)^2 (x + 2)^2 \)[/tex], counting multiplicities, is [tex]\( 4 \)[/tex].
