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Which statement describes the graph of this polynomial function?

[tex]\[ f(x) = x^5 - 6x^4 + 9x^3 \][/tex]

A. The graph crosses the x-axis at [tex]\( x=0 \)[/tex] and touches the x-axis at [tex]\( x=3 \)[/tex].
B. The graph touches the x-axis at [tex]\( x=0 \)[/tex] and crosses the x-axis at [tex]\( x=3 \)[/tex].
C. The graph crosses the x-axis at [tex]\( x=0 \)[/tex] and touches the x-axis at [tex]\( x=-3 \)[/tex].
D. The graph touches the x-axis at [tex]\( x=0 \)[/tex] and crosses the x-axis at [tex]\( x=-3 \)[/tex].

Sagot :

GinnyAnswer
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].
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