2. Which of the following is the correct expansion of $$ (2 x-3 y)^{4} ? $$ a. $$ 81 x^{4}+216 x^{3} y+216 x^{2} y^{2}+96 x y^{3}+16 y^{4} $$ b. $$ 81 x^{4}-216 x^{3} y+216 x^{2} y^{2}+96 x y^{3}+16 y^{4} $$ c. $$ 16 x^{4}-96 x^{3} y-216 x^{2} y^{2}-216 x y^{3}-81 y^{4} $$ d. $$ 16 x^{4}-96 x^{3} y+216 x^{2} y^{2}-216 x y^{3}+81 y^{4} $$

Question

UpStudy AI · Accepted Answer

We have the binomial expansion of $$(2x - 3y)^4$$ given by

$$
(2x - 3y)^4 = \sum_{k=0}^{4} \binom{4}{k}(2x)^{4-k}(-3y)^{k}.
$$

**Step 1.** For $$k=0$$:

$$
\binom{4}{0}(2x)^4(-3y)^0 = 1 \cdot 16x^4 \cdot 1 = 16x^4.
$$

**Step 2.** For $$k=1$$:

$$
\binom{4}{1}(2x)^3(-3y)^1 = 4 \cdot 8x^3 \cdot (-3y) = -96x^3y.
$$

**Step 3.** For $$k=2$$:

$$
\binom{4}{2}(2x)^2(-3y)^2 = 6 \cdot 4x^2 \cdot 9y^2 = 216x^2y^2.
$$

**Step 4.** For $$k=3$$:

$$
\binom{4}{3}(2x)^1(-3y)^3 = 4 \cdot 2x \cdot (-27y^3) = -216xy^3.
$$

**Step 5.** For $$k=4$$:

$$
\binom{4}{4}(2x)^0(-3y)^4 = 1 \cdot 1 \cdot 81y^4 = 81y^4.
$$

**Final Expansion.**

Combining all the terms we get

$$
(2x - 3y)^4 = 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4.
$$

Comparing with the options, we see that the correct expansion is given in option d.
The correct expansion of $$ (2x - 3y)^4 $$ is $$ 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4 $$.

Which of the following is the correct expansion of
a.
b.
c.
d.

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