To determine the nature of the intersections of the given polynomial function [tex]\( f(x) = x^5 - 6x^4 + 9x^3 \)[/tex] with the [tex]\( x \)[/tex]-axis, we need to follow these steps:
1. Find the roots of the polynomial.
2. Determine the multiplicity of each root.
3. Analyze how the graph interacts with the [tex]\( x \)[/tex]-axis based on the multiplicity of the roots.
### Step 1: Find the roots
First, set the polynomial equal to zero to find the roots:
[tex]\[ f(x) = x^5 - 6x^4 + 9x^3 = 0 \][/tex]
Factor out the common term [tex]\( x^3 \)[/tex]:
[tex]\[ x^3 (x^2 - 6x + 9) = 0 \][/tex]
Then, simplify within the parentheses:
[tex]\[ x^2 - 6x + 9 \][/tex]
Notice that this is a perfect square trinomial:
[tex]\[ x^2 - 6x + 9 = (x - 3)^2 \][/tex]
Now, the polynomial can be written as:
[tex]\[ x^3 (x - 3)^2 = 0 \][/tex]
### Step 2: Determine the multiplicity of each root
From the factorized form, we can see the roots and their multiplicities:
- [tex]\( x = 0 \)[/tex] with multiplicity 3 (from [tex]\( x^3 \)[/tex])
- [tex]\( x = 3 \)[/tex] with multiplicity 2 (from [tex]\( (x - 3)^2 \)[/tex])
### Step 3: Analyze the nature of each root
The multiplicity of a root gives us information on whether the graph crosses or touches the [tex]\( x \)[/tex]-axis at that root:
- If the multiplicity is odd, the graph crosses the [tex]\( x \)[/tex]-axis at the root.
- If the multiplicity is even, the graph touches but does not cross the [tex]\( x \)[/tex]-axis at the root.
Using these rules:
- For [tex]\( x = 0 \)[/tex] (multiplicity 3, which is odd), the graph crosses the [tex]\( x \)[/tex]-axis.
- For [tex]\( x = 3 \)[/tex] (multiplicity 2, which is even), the graph touches the [tex]\( x \)[/tex]-axis.
### Conclusion
From our analysis, the correct statement that describes the graph of the polynomial function [tex]\( f(x) = x^5 - 6x^4 + 9x^3 \)[/tex] is:
- The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x=0 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x=3 \)[/tex].
So, the correct answer is:
The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x=0 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x=3 \)[/tex].
