To determine at which root the graph of the function [tex]\( f(x) = (x + 4)^6 (x + 7)^5 \)[/tex] crosses the [tex]\( x \)[/tex]-axis, we must analyze the roots and their multiplicities.
1. Identify the roots of the function:
The roots are found by setting the function equal to zero, i.e., solving [tex]\( f(x) = 0 \)[/tex].
[tex]\( (x + 4)^6 = 0 \Rightarrow x = -4 \)[/tex]
[tex]\( (x + 7)^5 = 0 \Rightarrow x = -7 \)[/tex]
Thus, the roots are [tex]\( x = -4 \)[/tex] and [tex]\( x = -7 \)[/tex].
2. Determine the multiplicities of the roots:
- For [tex]\( x = -4 \)[/tex], the factor [tex]\( (x + 4)^6 \)[/tex] has a multiplicity of 6.
- For [tex]\( x = -7 \)[/tex], the factor [tex]\( (x + 7)^5 \)[/tex] has a multiplicity of 5.
3. Analyze the behavior of the graph at each root:
- A root with an even multiplicity means the graph just touches the [tex]\( x \)[/tex]-axis at this root and turns back (it doesn't cross the axis).
- A root with an odd multiplicity means the graph crosses the [tex]\( x \)[/tex]-axis at this root.
- For the root [tex]\( x = -4 \)[/tex] (even multiplicity of 6), the graph touches the [tex]\( x \)[/tex]-axis and turns back.
- For the root [tex]\( x = -7 \)[/tex] (odd multiplicity of 5), the graph crosses the [tex]\( x \)[/tex]-axis.
Therefore, the graph of [tex]\( f(x) \)[/tex] crosses the [tex]\( x \)[/tex]-axis at the root [tex]\( -7 \)[/tex].
Answer: [tex]\(-7\)[/tex]
