Creating a non-cubic polynomial

[rainyrabbit]

Would you please give one or two (or more, sorry hehehe) to help me?

How can you make up a polynomial that:
- crosses the x-axis at -2
- just touches the x-axis at 1 ("touch" as the graph y=x^2 would only touch, not cross, the x-axis)
- and is above the x-axis between -2 and 1.

I know there must be equations that are not cubic that satisfy these conditions. Oh and the graph contains no additional contact with the x-axis. How would you create the equation? Please help^^

 

[0rthodontist]

If the equation can't be cubic, then make it quintic (highest term x^5) so it will have the same end behavior.

Let's say that you have a polynomial that is cubic with these properties (do you?). Then can you predict what would happen if you multiply it by (for example) (x^2 + 1)?

 

[J77]

So you want a function $$f(x)$$ such that:

$$\frac{df}{dx}=0$$ and $$\frac{d^2f}{dx^2}<0$$ at $$x=1$$

and

$$f(-2)=0$$...

 

[rainyrabbit]

well very many thanks I give you. Sorry, I am only at precalculus and just at the basic level, so I don't really know what happens when you multiply an equation by (x^2 + 1), but as I tried doing that it worked! The graph of the quintic equation looks very similar to my original cubic equation lol^^

Oh and J77, I'm afraid I do not yet know calculus. Sorry hehehe...

EDIT: Oh I just figured out why. Adding the factor (x^2 + 1) does not add additional x-intercepts!

 

[HallsofIvy]

Would you please give one or two (or more, sorry hehehe) to help me?

How can you make up a polynomial that:
- crosses the x-axis at -2

So one factor is (x-(-2))= (x+2)

- just touches the x-axis at 1 ("touch" as the graph y=x^2 would only touch, not cross, the x-axis)

Okay, the parabola is tangent to the x-axis and another factor is
(x-1)²

- and is above the x-axis between -2 and 1.

(x+2)(x-1)² itself, at x= 0, is (0+2)(-1)²= 2 which is "above" the x-axis- i.e. positive. Since a polynomial is of one sign between places where it is 0 and (x+2)(x-1)² is 0 only at -2 and 1, it's positive between -2 and 1: it looks to me like that works.

I know there must be equations that are not cubic that satisfy these conditions. Oh and the graph contains no additional contact with the x-axis. How would you create the equation? Please help^^

Yes, there are "equation" (I assume you mean polygons) that are not cubic yet satisfy these equation. For example, multiply (x+2)(x-1)² by x²+1 which is always positive and, since x²+ 1 is never 0, does not cross or touch the x-axis at any other point. But why? Your original problem did not state that the polynomial must be not be cubic.