First, factorize the function f(x):
\(f(x) = 4x^7 + 40x^6 + 100x^5=4x^5(x^2+10x+25)=4x^5(x+5)^2\).
When f(x)=0, you get that
\(4x^5(x+5)^2=0,\\x=0 \text{ or } x=-5\).
This means that the graph intersects x-axis at point (0,0) and (-5,0). But:
1. since the degree at term \(x^5\) is odd, point (0,0) is x-intercept;
2. since the degree at term \((x+5)^2\) is even, point (-5,0) is tangent point (see attached graph of the function).
Answer: The graph crosses the x axis at x = 0 and touches the x axis at x = –5.
