Online assignments = confusion

[Alana456]

I'm doing an online course, and am finding it to be extremely difficult to do without having an actual teacher that you can sit with and a classroom full of others you can work with.

1. One of the questions is:
Determine a general equation for the quartic function f(x) as described:
f(x)<0 when x< -2, f(x)>0 when -2<x<3, and f(x)<0 when x>3, the zeros are -2, -1, and 3.

I've never worked with quartic functions or anything like this.
If anyone has some guidance that would be fantastic!

 

[Mentallic]

It's not much more than just an extension from quadratics/cubics.
The general form is $$y=ax^4+bx^3+cx^2+dx+e$$ but this won't help much in this case.
Using the fact that for a quartic in the form $$y=a(x-b)(x-c)(x-d)(x-e)$$, the roots are $$b, c, d, e$$ so since you know the roots are -2,-1 and 3, It can be written as $$y=(ax+b)(x+2)(x+1)(x-3)$$
Now just see if you can add anything from the other info given. Such as finding whether a is positive or negative (the coefficient of the highest power ($$x^4$$) is important such as how it makes quadratics concave up or concave down).

 

[HallsofIvy]

Because "f(x)>0 when -2<x<3", the graph of f does not cross the x-axis at x= -1. But x= -1 is a zero so the graph must be tangent to the axis there. That means x= -1 is a double root" the four[/b roots are -2, -1, -1, 3, and so gives you the fourth factor that Mentallic writes as "ax+ b".