# Sum of coefficients on polynomial

• Aug 15th 2012, 08:52 AM
DIOGYK
Sum of coefficients on polynomial
Hello.
There's this problem in my algebra book:
Problem 101. Imagine that the polynomial $(1+2x)^{200}$ is converted to the standard form (the sum of powers of x with numerical coefficients). What is the sum of all the coefficients?

My solution. The solution of this problem is $2^{200}$ I think, am I right?
• Aug 15th 2012, 09:04 AM
emakarov
Re: Sum of coefficients on polynomial
Quote:

Originally Posted by DIOGYK
My solution. The solution of this problem is $2^{200}$ I think, am I right?

Hint: the sum of coefficients of a polynomial P is P(1).
• Aug 15th 2012, 09:08 AM
DIOGYK
Re: Sum of coefficients on polynomial
Quote:

Originally Posted by emakarov
Hint: the sum of coefficients of a polynomial P is P(1).

So it's $3^{200}$?
• Aug 15th 2012, 09:20 AM
emakarov
Re: Sum of coefficients on polynomial
Yes. Do you agree?
• Aug 15th 2012, 09:24 AM
DIOGYK
Re: Sum of coefficients on polynomial
I don't know, because when I learned pascal's triangle, since it's about coefficients of polynomials, sum of the coefficients were always $2^n$, isn't that right? or I think I'm confused maybe.
• Aug 15th 2012, 09:42 AM
emakarov
Re: Sum of coefficients on polynomial
Quote:

Originally Posted by DIOGYK
I don't know, because when I learned pascal's triangle, since it's about coefficients of polynomials, sum of the coefficients were always $2^n$, isn't that right?

Binomial coefficients can be used to find the sum of polynomial coefficients, but each polynomial requires its own approach. You don't claim that the sum of coefficients of any polynomial is $2^n$, do you?

In this particular problem, it is not necessary to use binomial coefficients. Do you see that the sum of coefficients of a polynomial P is P(1)?
• Aug 15th 2012, 09:48 AM
richard1234
Re: Sum of coefficients on polynomial
Quote:

Originally Posted by DIOGYK
I don't know, because when I learned pascal's triangle, since it's about coefficients of polynomials, sum of the coefficients were always $2^n$, isn't that right? or I think I'm confused maybe.

Pascal's triangle works if you're expanding something like $(1+x)^n$. If you have something else, like $(1+2x)^n$, you need both binomial coefficients and the $2^k$ factor. Basically, you can't say that the sum of coefficients is always $2^n$.

On the other hand, P(1) gives you the sum of coefficients.
• Aug 15th 2012, 09:54 AM
DIOGYK
Re: Sum of coefficients on polynomial
Quote:

Originally Posted by richard1234
Pascal's triangle works if you're expanding something like $(1+x)^n$. If you have something else, like $(1+2x)^n$, you need both binomial coefficients and the $2^k$ factor. Basically, you can't say that the sum of coefficients is always $2^n$.

On the other hand, P(1) gives you the sum of coefficients.

I understand, Thanks for help.

Quote:

Originally Posted by emakarov
Binomial coefficients can be used to find the sum of polynomial coefficients, but each polynomial requires its own approach. You don't claim that the sum of coefficients of any polynomial is $2^n$, do you?

In this particular problem, it is not necessary to use binomial coefficients. Do you see that the sum of coefficients of a polynomial P is P(1)?

Thank you. I understand it now. And yes, I see that sum of coefficients of a polynomial is P(1). Thanks for help.

Also, is it okay if I create a new thread on every problem that I will be unable to solve?
• Aug 15th 2012, 09:56 AM
emakarov
Re: Sum of coefficients on polynomial
Quote:

Originally Posted by DIOGYK
Also, is it okay if I create a new thread on every problem that I will be unable to solve?

Yes; this is preferred way.