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

Sagot :

GinnyAnsweridn
To analyze the graph of the given polynomial function:

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

we need to determine its roots and their multiplicities, as they will tell us how the graph behaves at those points.

1. Find the roots:

Set the polynomial equal to zero:

[tex]\[ x^5 - 6x^4 + 9x^3 = 0 \][/tex]

Factor out the common term [tex]\( x^3 \)[/tex]:

[tex]\[ x^3(x^2 - 6x + 9) = 0 \][/tex]

This can be split into two factors:

[tex]\[ x^3 = 0 \][/tex]
and
[tex]\[ x^2 - 6x + 9 = 0 \][/tex]

2. Solve for each factor:

For [tex]\( x^3 = 0 \)[/tex]:

[tex]\[ x = 0 \][/tex]

For [tex]\( x^2 - 6x + 9 = 0 \)[/tex]:

This is a quadratic equation. We can solve it by factoring:

[tex]\[ (x - 3)^2 = 0 \][/tex]

This gives us:

[tex]\[ x = 3 \][/tex]

So, the roots of the polynomial are [tex]\( x = 0 \)[/tex] and [tex]\( x = 3 \)[/tex].

3. Determine the multiplicities:

- [tex]\( x = 0 \)[/tex] has a multiplicity of 3 (since [tex]\( x^3 = 0 \)[/tex]).
- [tex]\( x = 3 \)[/tex] has a multiplicity of 2 (since [tex]\( (x - 3)^2 = 0 \)[/tex]).

4. Analyze the behavior at each root:

- For a root with an odd multiplicity (like 3 for [tex]\( x = 0 \)[/tex]), the graph touches the x-axis and turns around, without crossing it.
- For a root with an even multiplicity (like 2 for [tex]\( x = 3 \)[/tex]), the graph touches the x-axis and turns around, without crossing it.

From this analysis, we conclude:

- The graph touches the x-axis at [tex]\( x = 0 \)[/tex].
- The graph touches the x-axis at [tex]\( x = 3 \)[/tex].

Given the options:

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

None of these statements are correct based on our analysis. The accurate description of the graph is as follows:

"The graph touches the x-axis at [tex]\( x = 0 \)[/tex] and touches the x-axis at [tex]\( x = 3 \)[/tex]."
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