Let's analyze the function [tex]\( f(x) = (x + 4)^6 (x + 7)^5 \)[/tex] to determine at which roots the graph crosses the x-axis.
1. Identify the Roots:
The roots of the function [tex]\( f(x) \)[/tex] are the values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 0 \)[/tex].
- For [tex]\( (x + 4)^6 = 0 \)[/tex], we solve [tex]\( x + 4 = 0 \)[/tex], giving [tex]\( x = -4 \)[/tex].
- For [tex]\( (x + 7)^5 = 0 \)[/tex], we solve [tex]\( x + 7 = 0 \)[/tex], giving [tex]\( x = -7 \)[/tex].
So, the roots of the function are [tex]\( x = -4 \)[/tex] and [tex]\( x = -7 \)[/tex].
2. Determine the Behavior at Each Root:
We need to understand how the graph behaves at these roots.
- The term [tex]\( (x + 4)^6 \)[/tex]: The exponent 6 is an even number. When a function contains a term raised to an even power, the graph touches the x-axis at the root but does not cross it. Hence, at [tex]\( x = -4 \)[/tex], the graph touches the x-axis but does not cross it.
- The term [tex]\( (x + 7)^5 \)[/tex]: The exponent 5 is an odd number. When a function contains a term raised to an odd power, the graph crosses the x-axis at the root. Hence, at [tex]\( x = -7 \)[/tex], the graph crosses the x-axis.
3. Conclusion:
Given the analysis, the root at which the graph of the function [tex]\( f(x) = (x + 4)^6 (x + 7)^5 \)[/tex] crosses the x-axis is [tex]\( x = -7 \)[/tex].
So the answer to the question "At which root does the graph of [tex]\( f(x) = (x + 4)^6 (x + 7)^5 \)[/tex] cross the x-axis?" is:
[tex]\[
-7
\][/tex]
