Expand (2x-y^2)^9
To expand (2x - y^2)^9, we can use the binomial theorem. The binomial theorem states that:
(a + b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + C(n,2)a^(n-2) b^2 + ... + C(n,n-1)a^1 b^(n-1) + C(n,n)a^0 b^n
where C(n, k) represents the binomial coefficient and can be calculated as C(n, k) = n! / (k! * (n - k)!)
In our case, a = 2x and b = -y^2, and n = 9. Therefore, we have:
(2x - y^2)^9 = C(9,0)(2x)^9 (-y^2)^0 + C(9,1)(2x)^8 (-y^2)^1 + C(9,2)(2x)^7 (-y^2)^2 + ... + C(9,7)(2x)^2 (-y^2)^7 + C(9,8)(2x)^1 (-y^2)^8 + C(9,9)(2x)^0 (-y^2)^9
Now, let's calculate each term in the expansion:
First term:
C(9,0)(2x)^9 (-y^2)^0 = 1 * (2x)^9 * 1 = (2x)^9
Second term:
C(9,1)(2x)^8 (-y^2)^1 = 9 * (2x)^8 * (-y^2)
Third term:
C(9,2)(2x)^7 (-y^2)^2 = 36 * (2x)^7 * (y^4)
Continuing this process for each term, we eventually get the fully expanded form of (2x - y^2)^9.
Note: The expansion will consist of terms with alternating signs due to the negative power in b = -y^2.
To expand the expression (2x - y^2)^9, we can use the binomial theorem. The binomial theorem states that for any two numbers a and b and a positive integer n, the expansion of (a + b)^n can be expressed as the sum of the terms formed by taking the binomial coefficient of each term and raising each term to the appropriate power.
In this case, a = 2x and b = -y^2, and we want to expand the expression to the power of 9. The binomial coefficient can be found using the formula:
C(n, r) = n! / (r! * (n - r)!),
where n is the power we're expanding to (9 in this case), and r is the power of the variable we're raising (from 0 to 9). The term (a + b)^n can be expanded term by term using this binomial coefficient.
Let's go ahead and expand (2x - y^2)^9 step-by-step:
Term 1: (2x)^9 * (-y^2)^0
The binomial coefficient is C(9, 0) = 9! / (0! * (9 - 0)!) = 1.
(2x)^9 * (-y^2)^0 = (2x)^9 = 2^9 * x^9 = 512x^9.
Term 2: (2x)^8 * (-y^2)^1
The binomial coefficient is C(9, 1) = 9! / (1! * (9 - 1)!) = 9.
(2x)^8 * (-y^2)^1 = 9 * (2x)^8 * (-y^2) = 9 * 2^8 * x^8 * (-y^2) = -2304x^8y^2.
Continuing this process for the remaining terms, we get:
Term 3: (2x)^7 * (-y^2)^2
The binomial coefficient is C(9, 2) = 9! / (2! * (9 - 2)!) = 36.
(2x)^7 * (-y^2)^2 = 36 * (2x)^7 * (y^2)^2 = 36 * 2^7 * x^7 * y^4 = 10368x^7y^4.
Term 4: (2x)^6 * (-y^2)^3
The binomial coefficient is C(9, 3) = 9! / (3! * (9 - 3)!) = 84.
(2x)^6 * (-y^2)^3 = 84 * (2x)^6 * (-y^2)^3 = 84 * 2^6 * x^6 * (-y^6) = -96768x^6y^6.
Term 5: (2x)^5 * (-y^2)^4
The binomial coefficient is C(9, 4) = 9! / (4! * (9 - 4)!) = 126.
(2x)^5 * (-y^2)^4 = 126 * (2x)^5 * (y^2)^4 = 126 * 2^5 * x^5 * y^8 = 20160x^5y^8.
Term 6: (2x)^4 * (-y^2)^5
The binomial coefficient is C(9, 5) = 9! / (5! * (9 - 5)!) = 126.
(2x)^4 * (-y^2)^5 = 126 * (2x)^4 * (-y^2)^5 = 126 * 2^4 * x^4 * (-y^10) = -20160x^4y^10.
Term 7: (2x)^3 * (-y^2)^6
The binomial coefficient is C(9, 6) = 9! / (6! * (9 - 6)!) = 84.
(2x)^3 * (-y^2)^6 = 84 * (2x)^3 * (y^2)^6 = 84 * 2^3 * x^3 * y^12 = 96768x^3y^12.
Term 8: (2x)^2 * (-y^2)^7
The binomial coefficient is C(9, 7) = 9! / (7! * (9 - 7)!) = 36.
(2x)^2 * (-y^2)^7 = 36 * (2x)^2 * (-y^2)^7 = 36 * 2^2 * x^2 * (-y^14) = -10368x^2y^14.
Term 9: (2x)^1 * (-y^2)^8
The binomial coefficient is C(9, 8) = 9! / (8! * (9 - 8)!) = 9.
(2x)^1 * (-y^2)^8 = 9 * (2x)^1 * (y^2)^8 = 9 * 2^1 * x^1 * y^16 = 2304xy^16.
Term 10: (2x)^0 * (-y^2)^9
The binomial coefficient is C(9, 9) = 9! / (9! * (9 - 9)!) = 1.
(2x)^0 * (-y^2)^9 = 1 * (y^2)^9 = y^18.
Now, let's sum up all the terms:
(2x - y^2)^9 = 512x^9 - 2304x^8y^2 + 10368x^7y^4 - 96768x^6y^6 + 20160x^5y^8 - 20160x^4y^10 + 96768x^3y^12 - 10368x^2y^14 + 2304xy^16 - y^18.
So, the expanded form of (2x - y^2)^9 is given by the above expression.