Question
For the expansion of (a − 3b
2
)
5
, determine the general term, and
the coefficient of a
3b
4
.
Answers
GPT 3.5
To expand (a - 3b^2)^5, we can use the Binomial Theorem. According to the Binomial Theorem, the expansion of (a - 3b^2)^n can be written as:
(a - 3b^2)^n = C(n, 0) * a^n * (-3b^2)^0 + C(n, 1) * a^(n-1) * (-3b^2)^1 + C(n, 2) * a^(n-2) * (-3b^2)^2 + ... + C(n, n-1) * a^1 * (-3b^2)^(n-1) + C(n, n) * a^0 * (-3b^2)^n
In this case, for the expansion of (a - 3b^2)^5, we have n = 5.
The general term of the expansion will have the form C(5, k) * a^(5-k) * (-3b^2)^k, where k is the power of (-3b^2).
To find the coefficient of a^3b^4, we need to find the term in the expansion where the powers of a and b in each term add up to 3 and 4 respectively.
We can set up the equation:
(5 - k) + 2k = 3
5 - k + 2k = 3
5 + k = 3
k = 3 - 5
k = -2
Since k cannot be a negative number, this means that there is no term with a^3b^4 in the expansion.
Therefore, the general term of the expansion is given by C(5, k) * a^(5-k) * (-3b^2)^k, and the coefficient of a^3b^4 is 0.
(a - 3b^2)^n = C(n, 0) * a^n * (-3b^2)^0 + C(n, 1) * a^(n-1) * (-3b^2)^1 + C(n, 2) * a^(n-2) * (-3b^2)^2 + ... + C(n, n-1) * a^1 * (-3b^2)^(n-1) + C(n, n) * a^0 * (-3b^2)^n
In this case, for the expansion of (a - 3b^2)^5, we have n = 5.
The general term of the expansion will have the form C(5, k) * a^(5-k) * (-3b^2)^k, where k is the power of (-3b^2).
To find the coefficient of a^3b^4, we need to find the term in the expansion where the powers of a and b in each term add up to 3 and 4 respectively.
We can set up the equation:
(5 - k) + 2k = 3
5 - k + 2k = 3
5 + k = 3
k = 3 - 5
k = -2
Since k cannot be a negative number, this means that there is no term with a^3b^4 in the expansion.
Therefore, the general term of the expansion is given by C(5, k) * a^(5-k) * (-3b^2)^k, and the coefficient of a^3b^4 is 0.