# Thread: Find Polynomial from properties

1. ## 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

2. 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:

$p(x)=x^2 (ax^2+bx+c)$

and that $x=2$, $y=2$ is a point of inflection means that $p(2)=2$ and $p''(2)=0$ (is a root of odd multiplicity)

CB

3. 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.

4. You could also use the fact that since $\langle 2,2\rangle$ is a point of inflection, you must have $p''(2)=0.$

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

From $p(0)=0$, you can immediately conclude that your $e=0.$ Thus, your polynomial becomes

$p(x)=ax^{4}+bx^{3}+cx^{2}+dx.$

Next, you set $p'(0)=0,$ which implies that

$(4ax^{3}+3bx^{2}+2cx+d)|_{x\to 0}=0.$

From this equation, you can see that $d=0.$

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.