Expand the binomial by using the binomial theorem.
-$$\left( x ^ { 5 } + y \right) ^ { 6 }$$
A) $x ^ { 30 } + y ^ { 6 }$
B) $x ^ { 30 } + 6 x ^ { 25 } y + 15 x ^ { 20 } y ^ { 2 } + 20 x ^ { 15 } y ^ { 3 } + 15 x ^ { 10 } y ^ { 4 } + 6 x ^ { 5 } y ^ { 5 } + y ^ { 6 }$
C) $x ^ { 11 } + 6 x ^ { 10 } y + 15 x ^ { 9 } y ^ { 2 } + 20 x ^ { 8 } y ^ { 3 } + 15 x ^ { 7 } y ^ { 4 } + 6 x ^ { 5 } y ^ { 5 } + y ^ { 6 }$
D) $x ^ { 6 } + 6 x ^ { 5 } y + 15 x ^ { 4 } y ^ { 2 } + 20 x ^ { 3 } y ^ { 3 } + 15 x ^ { 2 } y ^ { 4 } + 6 x y ^ { 5 } + y ^ { 6 }$

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 $(x^5 + y)^6$, we get the expansion as in option B, with each term reflecting the coefficients from Pascal's triangle and the appropriate powers of $x^5$ and $y$.

Expand the Binomial by Using the Binomial Theorem (x5+y)6\left( x ^ { 5 } + y \right) ^ { 6 }(x5+y)6

Expand the Binomial by Using the Binomial Theorem $\left( x ^ { 5 } + y \right) ^ { 6 }$