Question

In how many places does the graph of f(x) = x4 + 10x3 + 24x2 cross or touch the horizontal axis?

Answer 1

Answer: 3 times

Explanation:

Setting up the equation

If the function touches the horizontal axis, that means y=0, or f(x)=0 at that point(s).

So, we set our equation equal to 0

`0=x^4+10x^3+24x^2\\`

Factor out x^2

`0=x^2(x^2+10x+24)`

So, either
`x^2` or
`(x^2+10x+24)` must equal zero

Factoring

`x^2+10x+24=0\\(x+6)(x+4)=0\\`

x=-6 or x=-4

from before,
`x^2 = 0\\x=0`

So, f(x) equals zero at 3 places, when x=0, when x=-6, and when x=-4