Use the binomial theorem to expand the expression.
-$( 4 x + 1 ) ^ { 5 }$
A) $1024 x ^ { 5 } + 1$
B) $1024 x ^ { 5 } + 256 x ^ { 4 } + 64 x ^ { 3 } + 16 x ^ { 2 } + 4 x + 1$
C) $1024 x ^ { 5 } + 20 x ^ { 4 } + 160 x ^ { 3 } + 160 x ^ { 2 } + 20 x + 1$
D) $1024 x ^ { 5 } + 1280 x ^ { 4 } + 640 x ^ { 3 } + 160 x ^ { 2 } + 20 x + 1$

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 $(4x + 1)^5$, we calculate each term: $1024x^5$ from $4^5x^5$, $1280x^4$ from $5 \cdot 4^4x^4$, $640x^3$ from $\binom{5}{2} \cdot 4^3x^3$, $160x^2$ from $\binom{5}{3} \cdot 4^2x^2$, $20x$ from $\binom{5}{4} \cdot 4x$, and $1$ from $1^5$, leading to the correct expansion.

Use the Binomial Theorem to Expand the Expression (4x+1)5( 4 x + 1 ) ^ { 5 }(4x+1)5

Use the Binomial Theorem to Expand the Expression $( 4 x + 1 ) ^ { 5 }$