Expand a Binomial Raised to a Power
-$$( x + 3 ) ^ { 5 }$$
A) $x ^ { 5 } + 15 x ^ { 4 } + 90 x ^ { 3 } + 270 x ^ { 2 } + 405 x + 243$
B) $x ^ { 5 } + 15 x ^ { 4 } + 90 x ^ { 3 } + 270 x ^ { 2 } + 405 x + 3$
C) $x ^ { 5 } + 15 x ^ { 4 } + 180 x ^ { 3 } + 540 x ^ { 2 } + 405 x + 243$
D) $x ^ { 5 } + 15 x ^ { 4 } + 180 x ^ { 3 } + 540 x ^ { 2 } + 405 x + 3$

Question

Quizplus · Accepted Answer

The correct expansion is obtained using the binomial theorem, which involves calculating the binomial coefficients (also known as "n choose k") for each term. The coefficients for a binomial expansion of $(x + y)^n$ are given by the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the power and $k$ is the term's position in the expansion, starting from 0. For $(x + 3)^5$, the coefficients are calculated as follows: 1 (for $x^5$), 5 (for $x^4$), 10 (for $x^3$), 10 (for $x^2$), 5 (for $x^1$), and 1 (for the constant term), multiplied by $3^k$ for each term respectively, resulting in the expansion provided in option A.

Expand a Binomial Raised to a Power - (x+3)5( x + 3 ) ^ { 5 }(x+3)5

Expand a Binomial Raised to a Power
- $( x + 3 ) ^ { 5 }$