Expand the binomial by using the binomial theorem.
-$$\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }$$
A) $\frac { 1 } { 81 } v ^ { 4 } - \frac { 16 } { 27 } v ^ { 3 } w + \frac { 32 } { 3 } v ^ { 2 } w ^ { 2 } - \frac { 256 } { 3 } v w ^ { 3 } + 256 w ^ { 2 }$
B) $\frac { 1 } { 81 } v ^ { 4 } + 256 w ^ { 2 }$
C) $\frac { 1 } { 3 } v ^ { 4 } - 4 \mathrm { w } ^ { 4 }$
D) $\frac { 1 } { 81 } v ^ { 4 } - \frac { 256 } { 3 } v ^ { 3 } w + \frac { 32 } { 3 } v ^ { 2 } w ^ { 2 } - \frac { 16 } { 27 } v w ^ { 3 } + 256 w ^ { 2 }$

Question

Quizplus · Accepted Answer

The binomial theorem states that $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$. Applying this to $\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }$, we get the coefficients and powers as in option A, correctly applying the binomial coefficients and the powers of each term.

Expand the Binomial by Using the Binomial Theorem (13v−4w)4\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }(31​v−4w)4

Expand the Binomial by Using the Binomial Theorem $\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }$