Prove Equation Without Substitution: a0+a1x+a2x^2+...+anx^n=0

[inferi]

hi,

I have this equation that i need to prove i used numbers and worked but you have to prove it without substitution, here is the question.

let a1,a2,...,an be all real number with the property that:

a0+(a1/2)+(a2/3)+...(an/n+1)=0


prove that:

a0+a1x+a2x^2+...+anx^n=0

this question is consider a challenging problem.

so anyone can please help? thank you

 

[SiddharthM]

from rudin, this is actually one of the easier problems. But you didn't really state the problem fully: it's asking you to show there exists a x in (0,1) so that

a0+a1x+a2x^2+...+anx^n=0

You will need to use the Mean Value Theorem. If f'(x)=a0+a1x+a2x^2+...+anx^n, what is f(x)?

 

[inferi]

sorry i did not put the full question it is really the interval (0,1) but there is no f(x) in the question only equations are:
1- a0+(a1/2)+(a2/3)+...(an/n+1)=0

2- a0+a1x+a2x^2+...+anx^n=0

that's it and if you use the mean value theorem you are going only to get a0 when x=0 and the same second equation when x=1
so howto do it? hank for your help

 

[SiddharthM]

IF you put

f'(x)=a0+a1x+a2x^2+...+anx^n

WHAT MUST f(x) be?

Use the Mean value theorem on f(x).

I think you need to review the statement of the mean value theorem and the theory in general because it your last post indicates you don't understand it well. The MVT works for FUNCTIONS not simply equations.

The problem REQUIRES you to CONSTRUCT a polynomial function based on the constants a_0,..,a_n - it is here that you are required to be somewhat clever (if you've ever differentiated or integrated before the proper function you need to create it obvious) And from there use the two equations given to show that the function vanishes at 1 and at 0, and the rest is an application of the mean value theorem.

 

[therector24]

i started it three times with the mean value theorem (roll's, general mean value theorem,& integral mean value theorem), how ever, it didn't work so if you were in my case what will be your way of solving this question!

 

[inferi]

ok we have to integral
f'(x)=a0+a1x+a2x^2+...+anx^n
so we can apply the mean value theorem, and the integral of this equation is a0+(a1/2)+(a2/3)+...(an/n+1)
is that right?

 

[HallsofIvy]

No, it's not. The integral is the FUNCTION
$$c+ a_0 x+ \frac{a_1}{2} x^2+ \frac{a_2}{3} x^3+ \cdot\cdot\cdot + \frac{a_n}{n+1} x^{n+1}$$
where c can be any constant.

Apply the mean value theorem to THAT function.