To determine the zeroes of the polynomial function \( f(x)=3x^6+30x^5+75x^4 \) and their multiplicities, we need to follow a systematic approach.
### Step-by-Step Solution
1. Factorize the Polynomial:
First, identify any common factors in the terms of \( f(x) \).
[tex]\[
f(x) = 3x^6 + 30x^5 + 75x^4
\][/tex]
Notice that each term has a common factor of \( 3x^4 \):
[tex]\[
f(x) = 3x^4(x^2 + 10x + 25)
\][/tex]
2. Further Factorize:
Now, we need to factorize the quadratic \( x^2 + 10x + 25 \) within the expression:
[tex]\[
x^2 + 10x + 25 = (x + 5)^2
\][/tex]
Therefore, we can rewrite the polynomial as:
[tex]\[
f(x) = 3x^4(x + 5)^2
\][/tex]
3. Determine the Zeroes and Their Multiplicities:
The zeroes of the polynomial function \( f(x) \) are found by setting \( f(x) = 0 \):
[tex]\[
3x^4(x + 5)^2 = 0
\][/tex]
This equation is satisfied when either \( x^4 = 0 \) or \( (x + 5)^2 = 0 \).
- Solving \( x^4 = 0 \):
[tex]\[
x = 0
\][/tex]
Since the exponent 4 corresponds to a multiplicity of 4, the zero \( x = 0 \) has a multiplicity of 4.
- Solving \( (x + 5)^2 = 0 \):
[tex]\[
x + 5 = 0 \implies x = -5
\][/tex]
The exponent 2 indicates a multiplicity of 2, so the zero \( x = -5 \) has a multiplicity of 2.
### Summary of the Zeroes and Their Multiplicities:
- Zero at \( x = -5 \) with multiplicity 2
- Zero at \( x = 0 \) with multiplicity 4
Thus, the correct description of the zeroes of the graph of \( f(x) = 3x^6 + 30x^5 + 75x^4 \) is:
- \( -5 \) with multiplicity 2
- \( 0 \) with multiplicity 4
From the given options, the correct answer is:
[tex]\[
\boxed{-5 \text{ with multiplicity 2 and } 0 \text{ with multiplicity 4}}
\][/tex]
