Use the Binomial Theorem to expand and simplify the expression. $( x + 5 ) ^ { 6 }$
A) $x ^ { 6 } + 30 x ^ { 5 } + 500 x ^ { 4 } + 2500 x ^ { 3 } + 9375 x ^ { 2 } + 15625 x + 625$
B) $x ^ { 6 } + 500 x ^ { 5 } + 375 x ^ { 4 } + 3125 x ^ { 3 } + 9375 x ^ { 2 } + 18750 x + 15625$
C) $x ^ { 6 } + 30 x ^ { 5 } + 375 x ^ { 4 } + 2500 x ^ { 3 } + 9375 x ^ { 2 } + 18750 x + 15625$
D) $x ^ { 6 } + 30 x ^ { 5 } + 375 x ^ { 4 } + 2500 x ^ { 3 } + 9375 x ^ { 2 } - 18750 x + 625$
E) $x ^ { 6 } + 30 x ^ { 5 } + 25 x ^ { 4 } + 3125 x ^ { 3 } + 9375 x ^ { 2 } + 18750 x + 15625$

Question

Use the Binomial Theorem to expand and simplify the expression.   $( x + 5 ) ^ { 6 }$  
A) $x ^ { 6 } + 30 x ^ { 5 } + 500 x ^ { 4 } + 2500 x ^ { 3 } + 9375 x ^ { 2 } + 15625 x + 625$
B) $x ^ { 6 } + 500 x ^ { 5 } + 375 x ^ { 4 } + 3125 x ^ { 3 } + 9375 x ^ { 2 } + 18750 x + 15625$
C) $x ^ { 6 } + 30 x ^ { 5 } + 375 x ^ { 4 } + 2500 x ^ { 3 } + 9375 x ^ { 2 } + 18750 x + 15625$
D) $x ^ { 6 } + 30 x ^ { 5 } + 375 x ^ { 4 } + 2500 x ^ { 3 } + 9375 x ^ { 2 } - 18750 x + 625$
E) $x ^ { 6 } + 30 x ^ { 5 } + 25 x ^ { 4 } + 3125 x ^ { 3 } + 9375 x ^ { 2 } + 18750 x + 15625$

Quizplus · Accepted Answer

The correct expansion using the Binomial Theorem involves calculating each term using combinations and the powers of $x$ and $5$. The coefficients follow the pattern of the binomial coefficients from Pascal's triangle, and each term's coefficient is a product of a binomial coefficient and a power of 5. The correct expansion is $x^6 + 30x^5 + 375x^4 + 2500x^3 + 9375x^2 + 18750x + 15625$, which matches option C.

Use the Binomial Theorem to Expand and Simplify the Expression (x+5)6( x + 5 ) ^ { 6 }(x+5)6

Use the Binomial Theorem to Expand and Simplify the Expression $( x + 5 ) ^ { 6 }$