Use the Binomial Theorem to determine the coefficient of x^14 in the

expansion of � (3x^2 − 1/ 3)^ �16 . Express the coefficient as a fraction in its
lowest terms.

Please help me with this I'm getting nowhere. And please explain the logic behind! Thanks so much

To determine the coefficient of x^14 in the expansion of (3x^2 - 1/3)^16 using the Binomial Theorem, you need to follow these steps:

1. Understand the Binomial Theorem:
The Binomial Theorem is a useful formula for expanding expressions of the form (a + b)^n, where a, b, and n are constants. It allows us to find the coefficients of each term in the expansion.

2. Write down the formula for 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, r)a^(n-r) b^r + ... + C(n, n)a^0 b^n, where C(n, r) represents the binomial coefficient.

3. Determine the values of the variables in the formula:
In your case, a = 3x^2 and b = -1/3, while n = 16.

4. Identify the term that contains x^14:
To find the coefficient of x^14, you need to determine in which term of the expansion x^14 appears. Since x^14 = (3x^2)^(14/2) * (-1/3)^(16-14), it follows that x^14 appears in the term C(16, 7)(3x^2)^7 * (-1/3)^(16-7).

5. Calculate the binomial coefficient:
Using the formula for the binomial coefficient, C(n, r) = n! / (r!(n-r)!), you can compute C(16, 7) as follows:
C(16, 7) = 16! / (7!(16-7)!) = 16! / (7!9!) = (16 * 15 * 14 * 13 * 12 * 11 * 10) / (7 * 6 * 5 * 4 * 3 * 2 * 1).

6. Calculate each part of the term:
Now you can substitute the values into the term:
C(16, 7)(3x^2)^7 * (-1/3)^(16-7) = [(16 * 15 * 14 * 13 * 12 * 11 * 10) / (7 * 6 * 5 * 4 * 3 * 2 * 1)] * (3x^2)^7 * (-1/3)^9.

7. Simplify the term:
Simplify the term further by expanding (3x^2)^7 and (-1/3)^9, and then multiplying all the factors together.

8. Identify the coefficient of x^14 in the term:
Once you have simplified the term, identify the coefficient of x^14. It will be some expression involving x, numerical coefficients, and powers of x.

9. Express the coefficient as a fraction in lowest terms:
If the coefficient is a fraction, simplify it by dividing both the numerator and denominator by their greatest common divisor (GCD) to get the fraction in its lowest terms.

By following these steps, you will be able to determine the coefficient of x^14 in the expansion of (3x^2 - 1/3)^16 and express it as a fraction in its lowest terms.