James applies the Binomial Theorem for the expansion of (2x−3)^6 . Which of the following is the second term in the expansion?(1 point)
a. -2160x^4
b. -576x^5
c. 576x^5
d. 2160x^4
To find the second term in the expansion, we need to identify the term with the exponent of x equal to 5.
The Binomial Theorem states that for the expansion of (a + b)^n, the term from the binomial expansion is given by:
(nCk) * a^(n-k) * b^k
In this case, we have (2x - 3)^6, so a = 2x and b = -3.
We are looking for the term with k = 5.
Using the formulas from the Binomial Theorem, we have:
(6C5) * (2x)^(6-5) * (-3)^5
= 6 * 2x * 243
= 1458x
Therefore, the second term in the expansion is 1458x.
However, none of the given options match this answer. Therefore, none of the options provided are correct.
To find the second term in the expansion of (2x - 3)^6, we can use the Binomial Theorem formula, which states that each term in the expansion is given by:
nCr * a^(n-r) * b^r
Where:
n is the exponent (6 in this case)
r represents the term number (starting from 0)
nCr is the number of combinations or binomial coefficients, given by n! / (r! * (n-r)!)
a represents the first term (2x in this case)
b represents the second term (-3 in this case)
For the second term, r = 1.
Using this information, let's calculate the second term:
nCr = 6C1 = 6! / (1! * (6-1)!) = 6
a^(n-r) = (2x)^(6-1) = (2x)^5 = 32x^5
b^r = (-3)^1 = -3
Therefore, the second term in the expansion is:
6 * 32x^5 * (-3) = -576x^5
The correct answer is b. -576x^5.
To find the second term in the expansion of (2x-3)^6, we need to apply the Binomial Theorem. The Binomial Theorem states that the general term in the expansion of (a+b)^n is given by:
T(r+1) = C(n, r) * a^(n-r) * b^r
Where T(r+1) is the coefficient of the (r+1)-th term in the expansion, C(n, r) is the binomial coefficient, a is the first term in the binomial (2x in this case), b is the second term in the binomial (-3 in this case), n is the power to which the binomial is raised (6 in this case), and r is the term number.
So, for the second term (r = 1), we have:
T(2) = C(6, 1) * (2x)^(6-1) * (-3)^1
C(6, 1) = 6!/[(6-1)!*1!] = 6
So, the second term is:
T(2) = 6 * (2x)^5 * (-3)
T(2) = 6 * 32x^5 * (-3) = -576x^5
Therefore, the correct answer is (b) -576x^5.