Thread: Find Polynomial from properties

Hello,

I'm preparing for my final exams and I got stuck in a question concerning differentiation:

Let p(x) be a polynomial of degree four whose graph touches the x-axis at the origin and has a horizontal point of inflection at T= (2,2).
Determine p(x).

Well, I understand that the function must be of the form:

p(x) = ax^4 + bx^3 + cx^2 + dx + e

and that

p(0)=0
p'(0)=0
p(2)=2

but how do you know proceed?

I would be very gladful if someone could help me

Originally Posted by zimtaepfelchen

Hello,

I'm preparing for my final exams and I got stuck in a question concerning differentiation:

Let p(x) be a polynomial of degree four whose graph touches the x-axis at the origin and has a horizontal point of inflection at T= (2,2).
Determine p(x).

Well, I understand that the function must be of the form:

p(x) = ax^4 + bx^3 + cx^2 + dx + e

and that

p(0)=0
p'(0)=0
p(2)=2

but how do you know proceed?

I would be very gladful if someone could help me

Touches the x-axis at the origin means that the polynomial is of the form:



and that , is a point of inflection means that and (is a root of odd multiplicity)

CB

Originally Posted by zimtaepfelchen

Hello,

I'm preparing for my final exams and I got stuck in a question concerning differentiation:

Let p(x) be a polynomial of degree four whose graph touches the x-axis at the origin and has a horizontal point of inflection at T= (2,2).
Determine p(x).

Well, I understand that the function must be of the form:

p(x) = ax^4 + bx^3 + cx^2 + dx + e

and that

p(0)=0
p'(0)=0
p(2)=2

but how do you know proceed?

I would be very gladful if someone could help me

see that x=0, that means the equation has a multiplicity of 2 roots.

use what you know about multiplicity, and you shall find the answer to those unknown variables.

You could also use the fact that since is a point of inflection, you must have

Indeed, I would probably just bootstrap my way into the solution by using differentiation all the way.

From , you can immediately conclude that your Thus, your polynomial becomes



Next, you set which implies that



From this equation, you can see that

Rewrite your new polynomial, and continue plugging in the equations you have.

[EDIT]: This I post merely as an alternative to CB's perfectly valid approach.