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]."
