Can you give an example as to how The expansion of (x + y)n and (x - y)n are different?
on x-y expanded, the odd powers of y have negative coefficents.
For n odd, the sign of the y-ters differ: (+) for the first expansion, (-) for the second. (For n even, the signs are the same.)
Certainly! The expansion of (x + y)^n and (x - y)^n differ in terms of the signs of the terms in the expansion.
Let's first expand (x + y)^n using the binomial theorem:
(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 * y^(n-1) + C(n, n) * x^0 * y^n
Each term in the expansion has a coefficient represented by the binomial coefficient C(n, k) and consists of the variables x and y raised to different powers.
Now, let's expand (x - y)^n using the same binomial theorem:
(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 * (-y)^(n-1) + C(n, n) * x^0 * (-y)^n
In this expansion, each term also has a coefficient represented by the binomial coefficient C(n, k), but now the powers of y alternate between positive and negative due to the subtraction.
So, the main distinction between the two expansions is the sign of the terms involving y. In (x + y)^n, the terms involving y all have positive signs, while in (x - y)^n, the signs of the terms involving y alternate between positive and negative.
Of course! To compare the expansion of (x + y)n and (x - y)n, let's start by understanding what expansion means in this context.
The expansion of (x + y)n and (x - y)n refers to expanding the binomial expression raised to the power of n. The expansion can be obtained using the binomial theorem or by using a combination of properties such as the distributive property and the power rule.
Let's take an example to illustrate the difference between these two expansions:
Example: Expand (x + y)3 and (x - y)3
To expand (x + y)3, we'll need to apply 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-1) * x^1 * y^(n-1) + C(n,n) * x^0 * y^n
In this expression, C(n,r) denotes the binomial coefficient, which is the number of ways to choose r items from a set of n distinct items. It can be calculated using the formula C(n,r) = n! / (r! * (n - r)!)
So, let's expand (x + y)3 using the 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
Simplifying each term, we get:
(x + y)3 = x^3 + 3x^2y + 3xy^2 + y^3
Now, let's expand (x - y)3 using the same approach:
(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 each term, we get:
(x - y)3 = x^3 - 3x^2y + 3xy^2 - y^3
As you can see, the main difference between the two expansions is the alternate signs in front of the terms. In the expansion of (x + y)3, the signs are all positive, whereas in the expansion of (x - y)3, the signs alternate between positive and negative terms.
So, in summary, the expansions of (x + y)n and (x - y)n differ primarily due to the signs in front of the terms. The expansion of (x + y)n will have all positive signs, while the expansion of (x - y)n will have alternating signs.