Answer:
To find the roots of the function f(x) = (x+4)^6 (x+7)^5, we need to identify the values of x where the graph of the function crosses the x-axis. The x-axis is the horizontal line where y = 0.
To find the values of x when y = 0, we set the function f(x) equal to zero and solve for x. In this case, we have:
(x+4)^6 (x+7)^5 = 0
For the entire expression to equal zero, either (x+4)^6 = 0 or (x+7)^5 = 0, since multiplying anything by zero results in zero.
Now, let's solve each equation separately:
For (x+4)^6 = 0, we need to find the value of x that makes (x+4) equal to zero. By subtracting 4 from both sides of the equation, we find:
x+4 = 0
x = -4
Therefore, -4 is a root of the equation.
For (x+7)^5 = 0, we need to find the value of x that makes (x+7) equal to zero. By subtracting 7 from both sides of the equation, we find:
x+7 = 0
x = -7
Therefore, -7 is another root of the equation.
In conclusion, the graph of f(x) = (x+4)^6 (x+7)^5 crosses the x-axis at the roots -4 and -7. Thus, the correct answers are A: -7 and B: -4.
