Question

question 9 of 10
the binomial expansion of $(x - 2y)^3$ is ______.

a. $x^3 - 6x^2y + 12xy^2 - 8y^3$

b. $2^3 \left(x^3 - 3x^2y + 3xy^2 - y^3\
ight)$

c. $x^3 - 3x^2y + 3xy^2 - y^3$

d. $x^3 + 3x^2y + 3xy^2 + y^3

Explanation:

Step1: Recall binomial theorem

The binomial expansion of $(a - b)^n = \sum_{k=0}^n \binom{n}{k}a^{n-k}(-b)^k$. Here, $a=x$, $b=2y$, $n=3$.

Step2: Calculate each term

• $k=0$: $\binom{3}{0}x^3(-2y)^0 = 1 \cdot x^3 \cdot 1 = x^3$
• $k=1$: $\binom{3}{1}x^2(-2y)^1 = 3 \cdot x^2 \cdot (-2y) = -6x^2y$
• $k=2$: $\binom{3}{2}x^1(-2y)^2 = 3 \cdot x \cdot 4y^2 = 12xy^2$
• $k=3$: $\binom{3}{3}x^0(-2y)^3 = 1 \cdot 1 \cdot (-8y^3) = -8y^3$

Step3: Combine terms

Sum the terms: $x^3 - 6x^2y + 12xy^2 - 8y^3$

Answer:

A. $x^3 - 6x^2y + 12xy^2 - 8y^3$