- #1
Karnage1993
- 133
- 1
Homework Statement
Homework Equations
The Attempt at a Solution
I am really stuck, I have no clue how to even begin. For part B I tried changing the RHS to factorials but I was left at a dead end there.
Dick said:They gave you a nice clue for the first part. Expand (1+x)^n and (1+x)^m and multiply them. Then do the equal expression (1+x)^(n+m). Equate equal powers of x.
Dick said:Ok, then start simple. (1+x)^3 times (1+x)^3 is the same as (1+x)^6. Write the expansion of them in terms of binomial coefficients, not numbers. Now equate equal powers of x. It's a little tedious, but it's worth doing this if you really don't see what to do.
Dick said:(1+x)^3=C(3,0)+C(3,1)*x+C(3,2)*x^2+C(3,3)*x^3. I meant expand it like that. I know they are simple numbers but don't write 1, 6, 15 etc. Write C(6,0), C(6,1), C(6,2) etc. That's the only way you are going to see what's going on. And equating equal powers just means if 1+2x+3x^2=a+bx+cx^2 for all x then a=1, b=2 and c=3. That sort of thing.
Karnage1993 said:So you want me to expand (1+x)^3 times (1+x)^3 including the combinations?
If I did that, it would end up being (3C0)*(3C0 + 3C1 + 3C2 + 3C3) etc until I multiplied the other three terms in the first bracket. I still would not know how to simplify even that! What is 3C0 times 3C0 simplified into a combination? The only way I can do that is by getting the actual coefficient, 1, and multiplying it by 1.
Say with 3C2 times 3C2, what would that simplify into using the nCr ?
The Binomial Theorem is a formula used in algebra to expand binomials. It states that (x + y)^n = Σ(n, k)x^k y^(n-k), where n is a non-negative integer and k ranges from 0 to n.
The Binomial Theorem is often used in proofs to simplify expressions involving binomials. It allows us to expand binomials to any power and easily calculate their coefficients.
Yes, the Binomial Theorem can be used to solve equations by expanding binomials and setting them equal to a known value. This allows us to find the values of the variables in the equation.
The Binomial Theorem has many applications in mathematics, including in algebra, calculus, and probability. It is also used in other fields such as physics, engineering, and computer science.
While the Binomial Theorem is a powerful tool, it does have limitations. It can only be used for binomials, and the terms in the expansion can become very large for higher powers. Additionally, the Binomial Theorem does not work for non-integer powers.