To determine the zeros and their multiplicities for the function [tex]\(f(x) = 3x^6 + 30x^5 + 75x^4\)[/tex], let's follow these steps:
1. Find the Zeros of the Polynomial:
- First, factor the polynomial where possible.
[tex]\[ f(x) = 3x^6 + 30x^5 + 75x^4 \][/tex]
- Factor out the greatest common factor (GCF) from all terms:
[tex]\[ f(x) = 3x^4(x^2 + 10x + 25) \][/tex]
- Now, we need to factor the quadratic part, [tex]\( x^2 + 10x + 25 \)[/tex]:
[tex]\[ x^2 + 10x + 25 = (x + 5)^2 \][/tex]
- Substitute this back into our function:
[tex]\[ f(x) = 3x^4(x + 5)^2 \][/tex]
2. Identify the Zeros and Their Multiplicities:
- A zero occurs when [tex]\( f(x) = 0 \)[/tex].
[tex]\[ 3x^4(x + 5)^2 = 0 \][/tex]
- This equation is satisfied if any of the factors are zero:
[tex]\[ x^4 = 0 \quad \text{or} \quad (x + 5)^2 = 0 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ x^4 = 0 \quad \Rightarrow \quad x = 0 \quad (\text{multiplicity 4}) \][/tex]
- From [tex]\((x + 5)^2 = 0\)[/tex]:
[tex]\[ (x + 5)^2 = 0 \quad \Rightarrow \quad x + 5 = 0 \quad \Rightarrow \quad x = -5 \quad (\text{multiplicity 2}) \][/tex]
3. Conclusion:
- The zeros of the polynomial [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex] are at [tex]\( x = 0 \)[/tex] with multiplicity 4 and [tex]\( x = -5 \)[/tex] with multiplicity 2.
Therefore, the correct description of the zeros is:
- [tex]\(-5\)[/tex] with multiplicity 2
- [tex]\(0\)[/tex] with multiplicity 4
Hence, the correct answer is:
- [tex]\(-5\)[/tex] with multiplicity 2 and [tex]\(0\)[/tex] with multiplicity 4
