How to Derive Polynomials from Given Conditions?

[Mathman23]

Hi

This is the character equation for a polynomial of degree where $n \geq 0$

$$p(x) = a_0 x^{n} + a_{1} x^{n-1} + a_2 x ^{n-2} + \cdots + a_{n-1}x + a_{n}$$

I'm presented with the following assignment:

Two polynomials $\mathrm{p, q}$ where n = 3. These polynomials can derived using the following conditions:

$$\begin{array}{ccc} p(0) = -1 & q(1) = 3 & p(0) = q(0) \\ p'(0) =1 & q'(1) = -2 & p'(0) = q'(0) \end{array}$$

If was told by professor that these polynomials can we written in the following form.

$$p(x) = (a_{0} + a_{1})x^{n-1} + (a_{1})x^{n-2} + \cdots + (a_{n-1})x + a_{n}$$

How do I apply formula to my assignment??

Thanks in advance

Sincerely
Fred

 

[HallsofIvy]

Restating, you have two 3rd degree polynomials, p and q, such that:
p(0)= -1, p'(0)= 1, q(1)= 3, q'(1)= -2, q(0)= p(0)= -1, q'(0)= p'(0)= 1.

Okay, write p(x)= a₀+ a₁x+ a₂x²+ a₃x³ and
q(x)= b₀+ b₁x+ b₂x²+ b₃x³

p(0)= a₀= -1 and p'(0)= a₁= 1. Also, since q(0)= a(0), b₀= -1 and since q'(0)= p'(0)= 1, b₁= 1.

So far we have p(x)= -1+ x+ a₂x²+ a₃x³ and since that is all the information given about p(x), I don't see anyway of determining a₂ or a₃.

We have q(x)= -1+ x+ b₂x²+ b₃x³
so q(1)= -1+ 1+ b₂+ b₃= 3
and q'(x)= 1+ 2b₂2+ 3b₂ so
q'(1)= 1+ 2b₂+ 3b₂= -2.

You can solve those two linear equations for b₂ and b₃, completely determining q(x) but, unless you are given more information, there is no way to completely determine p(x).

That's assuming that assignment was to determine the two polynomials! You never did say what the assignment was.

 

[Mathman23]

Hi

I'm told that the resulting two polynomials are:

I'm told that the resulting two polynomials of degree 3 are:

p(x) = (2 + s - 2t) x^3 + (3 + 2s - 3t) x^2 + s*x + t

q(x) = (-6 + s + 2t) x^3 + (9 - 2s - 3t) x^2 + s*x +t

where s,t belong to R.

But my textbook doesn't have any information on how one can arrive at the above result.

Maybe You Hall can provide me with a hint on howto obtain the above result ?

Sincerley and Best Regards,

Fred

Restating, you have two 3rd degree polynomials, p and q, such that:
p(0)= -1, p'(0)= 1, q(1)= 3, q'(1)= -2, q(0)= p(0)= -1, q'(0)= p'(0)= 1.

Okay, write p(x)= a₀+ a₁x+ a₂x²+ a₃x³ and
q(x)= b₀+ b₁x+ b₂x²+ b₃x³

p(0)= a₀= -1 and p'(0)= a₁= 1. Also, since q(0)= a(0), b₀= -1 and since q'(0)= p'(0)= 1, b₁= 1.

So far we have p(x)= -1+ x+ a₂x²+ a₃x³ and since that is all the information given about p(x), I don't see anyway of determining a₂ or a₃.

We have q(x)= -1+ x+ b₂x²+ b₃x³
so q(1)= -1+ 1+ b₂+ b₃= 3
and q'(x)= 1+ 2b₂2+ 3b₂ so
q'(1)= 1+ 2b₂+ 3b₂= -2.

You can solve those two linear equations for b₂ and b₃, completely determining q(x) but, unless you are given more information, there is no way to completely determine p(x).

That's assuming that assignment was to determine the two polynomials! You never did say what the assignment was.