Question

I've been working on this problem for a week and I still am stuck on what to do next.

Answer 1

\[Factor f(x)= x^4 + x^3 - 8x^2 + 6 + 36 completely, then sketch the graph.\]

Answer 2

I've factored it completely to \[(x+2) (x+3) (x^2 - 4x + 6)\]

Answer 3

How do I go about finding points to graph it?

Answer 4

You know two points where it crosses the x-axis. So you can plot those two points.
The leading term is x^4 which is an even power and so the ends of the graph will behave much like x^2.
As x-> +/- infinity, y-> +infinity.
If you take the derivative, it will be a 3rd degree polynomial and when equated to zero it is likely to have 3 maximum/minimum points.
Find those points.
Then for the rest of it you can put a few values of x such as -3, -2, -1, 0, 1, 2, 3 and calculate the y-values.
That should give sufficient info to draw the graph.

Answer 5

You'll need much more than that to graph it.

If you're 100% the factoring is correct, the will notice that the quadratic form has two complex roots which cannot be plotted on your real plane.

To graph it would require some effort. You have the two real roots (-2 and -3). Those are the points where the graph goes through the X axis. For x=0 you'll get the the point at which the graph goes through the Y axis.

You'll have to apply the first derivative for the possible max/min global/local points. A second derivative to that in order to at least get the inflections. Lim of f(x) as x goes to infinity and - infinity. Some asymptotes would be in order but not strictly necessary.

Answer 6

You both are great, thank you for taking the time to help me! Unfortunately, I still have no idea.. Thank you, though!

Answer 7

You have not taken calculus yet?