Expand (2x-y^2)^9 by finding the term in x^7y^5
To find the term in x^7y^5 when expanding (2x-y^2)^9, we need to choose one term from each of the binomials 2x and -y^2 that will multiply to give x^7y^5.
We can choose x^7 from 2x and y^5 from -y^2.
Using the binomial theorem, we can write the term as:
C(9, 7) * (2x)^7 * (-y^2)^5
C(9, 7) represents the combination of 9 choose 7 and can be calculated as follows:
C(9, 7) = 9! / (7! * (9-7)!) = 9! / (7! * 2!) = (9 * 8) / (2 * 1) = 36
Substituting this value, the term becomes:
36 * (2x)^7 * (-y^2)^5
Simplifying further:
36 * 2^7 * x^7 * (-1)^5 * y^10
36 * 128 * x^7 * (-1) * y^10
4608 * x^7 * (-y^10)
Therefore, the term in x^7y^5 when expanding (2x-y^2)^9 is -4608x^7y^10.
To expand the expression (2x - y^2)^9 and find the term in x^7y^5, we can use the binomial theorem.
According to the binomial theorem, the term in the expansion of (2x - y^2)^9 that contains x^7y^5 is given by:
T(n + 1) = (9Cn) * (2x)^(9 - n) * (-y^2)^n
where T(n + 1) represents the term number (n + 1) in the expansion and n represents the number of occurrences of (2x) in that term.
In this case, we want to find the term with x^7y^5, so we need n = 7.
T(7 + 1) = (9C7) * (2x)^(9 - 7) * (-y^2)^7
Simplifying further, we get:
T(8) = (9C7) * (2x)^2 * (-y^2)^7
Now, let's calculate the values of (9C7), (2x)^2, and (-y^2)^7:
(9C7) = 9! / (7! * (9 - 7)!) = 36
(2x)^2 = 4x^2
(-y^2)^7 = -(y^2)^7 = -y^14
Substituting these values back into the equation, we have:
T(8) = (9C7) * (2x)^2 * (-y^2)^7
= 36 * 4x^2 * (-y^14)
= -144x^2y^14
Therefore, the term in x^7y^5 in the expansion of (2x - y^2)^9 is -144x^2y^14.