To find the zeroes of the polynomial function [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex] and their multiplicities, we will follow these steps:
1. Factor the polynomial: Start by factoring out the greatest common factor (GCF) from the terms.
2. Solve for the roots: Find the values of [tex]\( x \)[/tex] that make each factor equal to zero.
3. Determine multiplicities: Identify how many times each root appears.
### Step 1: Factor the Polynomial
First, notice that each term in [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex] has a common factor of [tex]\( 3x^4 \)[/tex]. Let's factor this out:
[tex]\[ f(x) = 3x^4(x^2 + 10x + 25) \][/tex]
### Step 2: Solve for the Roots
Next, we solve for the roots of each factor separately:
Factor 1: [tex]\( 3x^4 \)[/tex]
The roots of [tex]\( 3x^4 = 0 \)[/tex] are:
[tex]\[ 3x^4 = 0 \][/tex]
[tex]\[ x^4 = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
Since [tex]\( x = 0 \)[/tex] is raised to the fourth power, it has a multiplicity of 4.
Factor 2: [tex]\( x^2 + 10x + 25 \)[/tex]
The roots of the quadratic [tex]\( x^2 + 10x + 25 = 0 \)[/tex] can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = 25 \)[/tex].
[tex]\[ x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 25}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-10 \pm \sqrt{100 - 100}}{2} \][/tex]
[tex]\[ x = \frac{-10 \pm \sqrt{0}}{2} \][/tex]
[tex]\[ x = \frac{-10 \pm 0}{2} \][/tex]
[tex]\[ x = \frac{-10}{2} \][/tex]
[tex]\[ x = -5 \][/tex]
The root [tex]\( x = -5 \)[/tex] appears once for each instance in the quadratic factor, and since the quadratic can be written as [tex]\((x + 5)^2\)[/tex], the root [tex]\( x = -5 \)[/tex] has a multiplicity of 2.
### Step 3: Identify Zeroes and Their Multiplicities
Summarizing the zeroes and their multiplicities:
- The root [tex]\( x = 0 \)[/tex] has a multiplicity of 4.
- The root [tex]\( x = -5 \)[/tex] has a multiplicity of 2.
### Final Answer
The zeroes of the graph of the function [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex] are:
[tex]\[ -5 \text{ with multiplicity 2} \text{ and } 0 \text{ with multiplicity 4} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{-5 \text{ with multiplicity 2 and } 0 \text{ with multiplicity 4}} \][/tex]
