Use the binomial theorem to expand the binomial. $$\left( \frac { a } { 4 } - b \right) ^ { 5 }$$
A) $\frac { 1 } { 1024 } a ^ { 5 } - \frac { 5 } { 256 } a ^ { 4 } b + \frac { 5 } { 32 } a ^ { 3 } b ^ { 2 } - \frac { 5 } { 16 } a ^ { 3 } b ^ { 3 } + \frac { 5 } { 8 } a ^ { 2 } b ^ { 3 } - \frac { 5 } { 4 } a b ^ { 4 } + b ^ { 5 }$
B) $\frac { 1 } { 4 } a ^ { 5 } - \frac { 5 } { 4 } a ^ { 4 } b + \frac { 5 } { 2 } a ^ { 3 } b ^ { 2 } - \frac { 5 } { 2 } a ^ { 2 } b ^ { 3 } - \frac { 5 } { 4 } a b ^ { 4 } - b ^ { 5 }$
C) $\frac { 1 } { 1024 } a ^ { 5 } - \frac { 5 } { 256 } a ^ { 4 } b + \frac { 5 } { 32 } a ^ { 3 } b ^ { 2 } - \frac { 5 } { 8 } a ^ { 2 } b ^ { 3 } + \frac { 5 } { 4 } a b ^ { 4 } - b ^ { 5 }$
D) $\frac { 1 } { 4 } a ^ { 5 } - \frac { 5 } { 4 } a ^ { 4 } b + \frac { 5 } { 2 } a ^ { 3 } b ^ { 2 } - 5 a ^ { 3 } b ^ { 3 } + \frac { 5 } { 2 } a ^ { 2 } b ^ { 3 } - \frac { 5 } { 4 } a b ^ { 4 } + b ^ { 5 }$

Question

Quizplus · Accepted Answer

The correct expansion using the binomial theorem involves calculating each term by combining the coefficients from the binomial coefficient $\binom{n}{k}$, where $n$ is the power of the binomial (in this case, 5) and $k$ is the term number, with the powers of each part of the binomial $\left(\frac{a}{4}\right)^{n-k}(-b)^k$. The correct coefficients and powers of $a$ and $b$ are found in option C, taking into account the division by $4^k$ for the $a$ terms due to the $\frac{a}{4}$ part of the binomial.

Use the Binomial Theorem to Expand the Binomial (a4−b)5\left( \frac { a } { 4 } - b \right) ^ { 5 }(4a​−b)5

Use the Binomial Theorem to Expand the Binomial $\left( \frac { a } { 4 } - b \right) ^ { 5 }$