Use binomial the porem to expand (x+y)25
Did you mean (x+y)^25 ?
Do you really expect me to write out all 26 terms ?
What is "the porem" ?
That's "theorem" with an uncorrected typo.
Type in (x+y)^25 at wolframalpha.com to see the whole grisly expansion.
If I were a teacher, I'd accept
25
Ī£ 25Ck x^(25-k) y^k
k=0
since it would indicate that you know about binomial coefficients and their use in the the the porem. :-)
To expand (x+y)^25 using the binomial theorem, we can use the formula:
(x+y)^n = C(n,0)*x^n*y^0 + C(n,1)*x^(n-1)*y^1 + C(n,2)*x^(n-2)*y^2 + ... + C(n,r)*x^(n-r)*y^r + ... + C(n,n)*x^0*y^n
Where C(n,r) represents the binomial coefficient, given by:
C(n,r) = n! / (r! * (n-r)!)
Let's substitute the values into the formula:
First term:
C(25,0)*x^25*y^0 = 1*x^25*y^0 = x^25
Second term:
C(25,1)*x^24*y^1 = 25*x^24*y^1 = 25*x^24*y
Third term:
C(25,2)*x^23*y^2 = (25*24/2)*x^23*y^2 = 300*x^23*y^2
And so on, until we reach the last term:
C(25,25)*x^0*y^25 = 1*x^0*y^25 = y^25
Therefore, the expanded form of (x+y)^25 is:
x^25 + 25*x^24*y + 300*x^23*y^2 + ... + 25*x*y^24 + y^25
To expand the binomial (x+y)^25, we can use the binomial theorem, which states that for any positive integer n:
(x+y)^n = C(n,0)*x^n*y^0 + C(n,1)*x^(n-1)*y^1 + C(n,2)*x^(n-2)*y^2 + ... + C(n,n)*x^0*y^n
where C(n,r) represents the binomial coefficient, which is the number of ways to choose r items from a set of n items.
To find the expansion of (x+y)^25, we need to identify the values of C(n,r) for all r from 0 to 25. The binomial coefficients can be calculated using Pascal's Triangle or using the formula C(n,r) = n!/(r!(n-r)!), where "!" denotes factorial.
Let's calculate the expansion step by step:
Step 1: Calculate the binomial coefficients:
C(25,0) = 25! / (0! * (25-0)!) = 1
C(25,1) = 25! / (1! * (25-1)!) = 25
C(25,2) = 25! / (2! * (25-2)!) = 300
...
C(25,23) = 25! / (23! * (25-23)!) = 300
C(25,24) = 25! / (24! * (25-24)!) = 25
C(25,25) = 25! / (25! * (25-25)!) = 1
Step 2: Substitute the calculated binomial coefficients into the binomial expansion formula:
(x+y)^25 = C(25,0)*x^25*y^0 + C(25,1)*x^24*y^1 + C(25,2)*x^23*y^2 + ... + C(25,23)*x^2*y^23 + C(25,24)*x^1*y^24 + C(25,25)*x^0*y^25
Simplifying the terms, we have:
(x+y)^25 = x^25 + 25x^24*y + 300x^23*y^2 + ... + 300x^2*y^23 + 25x*y^24 + y^25
So, the expansion of (x+y)^25 using the binomial theorem is:
x^25 + 25x^24*y + 300x^23*y^2 + ... + 300x^2*y^23 + 25x*y^24 + y^25