Question

Which statement describes the graph of f(x) = 4x7 + 40x6 + 100x5?

The graph crosses the x axis at x = 0 and touches the x axis at x = 5.
The graph touches the x axis at x = 0 and crosses the x axis at x = 5.
The graph crosses the x axis at x = 0 and touches the x axis at x = –5.
The graph touches the x axis at x = 0 and crosses the x axis at x = –5.

Answer 1

The answer is C, plain and simple.

Answer 2

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.