9. Find the fourth term in the expansion of

Question

9. Find the fourth term in the expansion of

(3𝑥 − 𝑦)6
A. −540𝑥3𝑦3
B. −540𝑥4𝑦2
C. 540𝑥3𝑦3
D. −27𝑥3𝑦3

Answer 1

Using the binomial theorem, we know that the fourth term in the expansion of $(3𝑥 − 𝑦)^6$ will be:

$\binom{6}{3}(3𝑥)^3(-𝑦)^3 = 20\times27𝑥^3\times(-𝑦)^3 = -540𝑥^3𝑦^3$

Therefore, the answer is $\boxed{\text{A. }-540𝑥^3𝑦^3}$.

Answer 2

To find the fourth term in the expansion of (3𝑥 − 𝑦)6, we can use the binomial theorem. The binomial theorem states that the coefficient of the 𝑥^𝑎𝑦^𝑏 term in the expansion of (𝑎𝑥 + 𝑏𝑦)^𝑛 is given by the formula:

(𝑛𝐶𝑎)(𝑎𝑥)^𝑎(𝑏𝑦)^𝑏

In this case, we are looking for the fourth term, so 𝑎 = 2 and 𝑏 = 4. Plugging these values into the formula, we get:

(6𝐶2)(3𝑥)^2(−𝑦)^4
= (6! / (2!(6-2)!)) * (3^2𝑥^2)(−𝑦)^4
= 15 * 9𝑥^2𝑦^4
= 135𝑥^2𝑦^4

So, the fourth term in the expansion of (3𝑥 − 𝑦)6 is 135𝑥^2𝑦^4.

Therefore, the correct answer is option D: −27𝑥3𝑦3.