Use the Binomial Theorem to expand and simplify the expression. $( 5 x + 2 y ) ^ { 4 }$
A) $625 x ^ { 4 } + 1000 x ^ { 3 } y + 600 x ^ { 2 } y ^ { 2 } + 160 x y ^ { 3 } + 16 y ^ { 4 }$
B) $625 x ^ { 4 } + 250 x ^ { 3 } y + 100 x ^ { 2 } y ^ { 2 } + 40 x y ^ { 3 } + 16 y ^ { 4 }$
C) $625 x ^ { 4 } + 4 x ^ { 3 } y + 6 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 } + y ^ { 4 }$
D) $125 x ^ { 4 } + 200 x ^ { 3 } y + 120 x ^ { 2 } y ^ { 2 } + 32 x y ^ { 3 } + 16 y ^ { 4 }$
E) $125 x ^ { 3 } + 150 x ^ { 2 } y + 60 x y ^ { 2 } + 8 y ^ { 3 }$

Question

Quizplus · Accepted Answer

The correct expansion using the Binomial Theorem for $(5x + 2y)^4$ involves calculating each term by combining the coefficients from the binomial expansion formula, $\binom{n}{k} a^{n-k}b^k$, where $n=4$ in this case, and $a=5x$, $b=2y$. The coefficients follow the pattern of Pascal's triangle, and each term's coefficients are calculated accordingly, leading to the correct expansion as given in option A.

Use the Binomial Theorem to Expand and Simplify the Expression (5x+2y)4( 5 x + 2 y ) ^ { 4 }(5x+2y)4

Use the Binomial Theorem to Expand and Simplify the Expression $( 5 x + 2 y ) ^ { 4 }$