Use the binomial theorem to expand the binomial. $$\left( \frac { a } { 2 } + t \right) ^ { 4 }$$
A) $\quad a ^ { 4 } + a ^ { 3 } t + a ^ { 2 } t ^ { 2 } + a t ^ { 3 } + t ^ { 4 }$
B) $\quad \frac { 1 } { 8 } a ^ { 4 } + \frac { 1 } { 2 } a ^ { 3 } t + \frac { 3 } { 2 } a ^ { 2 } t ^ { 2 } + 2 a t ^ { 3 } + t ^ { 4 }$
C) $\quad \frac { 1 } { 2 } a ^ { 4 } + t ^ { 4 }$
D) $\quad \frac { 1 } { 16 } a ^ { 4 } + \frac { 1 } { 2 } a ^ { 3 } t + \frac { 3 } { 2 } a ^ { 2 } t ^ { 2 } + 2 a t ^ { 3 } + t ^ { 4 }$
E) $\quad \frac { 1 } { 16 } a ^ { 4 } + \frac { 1 } { 8 } a ^ { 3 } t + \frac { 1 } { 4 } a ^ { 2 } t ^ { 2 } + \frac { 1 } { 2 } a t ^ { 3 } + t ^ { 4 }$

Question

Quizplus · Accepted Answer

Using the binomial theorem, we have
$$\left( \frac { a } { 2 } + t \right) ^ { 4 } = \binom{4}{0}\left(\frac{a}{2}\right)^4 t^0 + \binom{4}{1}\left(\frac{a}{2}\right)^3 t^1 + \binom{4}{2}\left(\frac{a}{2}\right)^2 t^2 + \binom{4}{3}\left(\frac{a}{2}\right)^1 t^3 + \binom{4}{4}\left(\frac{a}{2}\right)^0 t^4$$
Simplifying this expression gives us
$$ \frac {1}{16} a^{4} + \frac{1}{2} a^{3} t + \frac{3}{4} a^{2} t^{2} + \frac{1}{2}a t^{3} + t^{4}$$
Therefore, the answer is D.

Use the Binomial Theorem to Expand the Binomial (a2+t)4\left( \frac { a } { 2 } + t \right) ^ { 4 }(2a​+t)4

Use the Binomial Theorem to Expand the Binomial $\left( \frac { a } { 2 } + t \right) ^ { 4 }$