How does the expansion of (x + y)n and (x - y)n differ? If you can offer an example that would be very helpful.
Thank you
think about it. (x+y)^n has + signs for all the terms.
(x-y)^n has minus signs for all the odd powers of n, since that is the same as
(x + (-y))^n
I'm sure you can provide your own examples. You might also try this web site:
http://www.wolframalpha.com/input/?i=(x-y)%5E5
Just scroll down past the graphs.
The expansion of (x + y)^n and (x - y)^n follows different patterns. Let's take a closer look at each expansion:
Expansion of (x + y)^n:
The binomial theorem states that the expansion of (x + y)^n can be obtained by using 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, n-1)x^1 y^(n-1) + C(n, n)x^0 y^n
Here, C(n, r) represents the binomial coefficient, which calculates the number of ways you can choose 'r' items from a set of 'n' items. The binomial coefficient is given by the formula C(n, r) = n! / (r! * (n-r)!), where '!' denotes the factorial of a number.
Expansion of (x - y)^n:
The expansion of (x - y)^n follows a similar pattern as (x + y)^n, but with alternating signs. This can be obtained using 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, n-1)x^1 (-y)^(n-1) + C(n, n)x^0 (-y)^n
By substituting the values of C(n, r) and simplifying the algebraic expressions, you can expand both (x + y)^n and (x - y)^n.
Let's consider an example to illustrate the difference:
Example:
Let's expand (x + y)^3 and (x - y)^3:
Expansion of (x + y)^3:
Using the expansion formula, we have:
(x + y)^3 = C(3, 0)x^3 y^0 + C(3, 1)x^2 y^1 + C(3, 2)x^1 y^2 + C(3, 3)x^0 y^3
Simplifying the terms, this becomes:
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
Expansion of (x - y)^3:
Similarly, using the expansion formula:
(x - y)^3 = C(3, 0)x^3 (-y)^0 + C(3, 1)x^2 (-y)^1 + C(3, 2)x^1 (-y)^2 + C(3, 3)x^0 (-y)^3
Again, simplifying the terms, this becomes:
(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
As you can see, the only difference between the two expansions is the alternating signs in the terms when expanding (x - y)^n. This difference arises due to the negative sign in the expansion formula for (x - y)^n.
I hope this explanation helps you understand the difference between the expansions of (x + y)^n and (x - y)^n.