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Expand (1-2p)^10 in binomial expansion

User Benderto
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5 votes

Answer:

Here is the answer.

Step-by-step explanation:

To expand the expression (1-2p)^10 using the binomial expansion, we can apply the binomial theorem. The binomial theorem 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

where C(n, k) represents the binomial coefficient, given by C(n, k) = n! / (k! * (n-k)!).

In this case, we have (1-2p) as x and we want to expand it to the power of 10. Let's calculate the expansion:

(1-2p)^10 = C(10, 0) * (1)^10 * (-2p)^0 + C(10, 1) * (1)^9 * (-2p)^1 + C(10, 2) * (1)^8 * (-2p)^2 + ... + C(10, 9) * (1)^1 * (-2p)^9 + C(10, 10) * (1)^0 * (-2p)^10

Simplifying further:

(1-2p)^10 = 1 * 1 * 1 + C(10, 1) * 1 * (-2p) + C(10, 2) * 1 * (4p^2) + ... + C(10, 9) * 1 * (-512p^9) + C(10, 10) * 1 * (1024p^10)

Now we can calculate the binomial coefficients and simplify the expression:

(1-2p)^10 = 1 - 20p + 180p^2 - 960p^3 + 3360p^4 - 8064p^5 + 13312p^6 - 15360p^7 + 11520p^8 - 5120p^9 + 1024p^10

Therefore, the expansion of (1-2p)^10 using the binomial theorem is 1 - 20p + 180p^2 - 960p^3 + 3360p^4 - 8064p^5 + 13312p^6 - 15360p^7 + 11520p^8 - 5120p^9 + 1024p^10.

User Jawad Ahbab
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8.5k points
6 votes

Final answer:

To expand (1-2p)^10 using the binomial expansion, we can use the formula (a + b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + ... + C(n,n-1)a^1 b^(n-1) + C(n,n)a^0 b^n, where C(n,k) = n! / (k! * (n-k)!). The expansion of (1-2p)^10 is 1 - 20p + 180p^2 - 960p^3 + 3360p^4 - 8064p^5 + 13440p^6 - 15360p^7 + 11520p^8 - 5120p^9 + 1024p^10.

Step-by-step explanation:

To expand (1-2p)^10 using the binomial expansion, we can use the formula (a + b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + ... + C(n,n-1)a^1 b^(n-1) + C(n,n)a^0 b^n, where C(n,k) = n! / (k! * (n-k)!).

In this case, a = 1, b = -2p, and n = 10.

Using the formula, we can find each term in the expansion:

Term 1: C(10,0) * 1^10 * (-2p)^0 = 1

Term 2: C(10,1) * 1^9 * (-2p)^1 = -20p

Term 11: C(10,10) * 1^0 * (-2p)^10 = (-2p)^10

Therefore, the expansion of (1-2p)^10 is 1 - 20p + 180p^2 - 960p^3 + 3360p^4 - 8064p^5 + 13440p^6 - 15360p^7 + 11520p^8 - 5120p^9 + 1024p^10.

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