Use Pascal's triangle to expand the binomial.
-$$\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 5 }$$
A) $x ^ { 15 } - 10 x ^ { 7 } y ^ { 2 } + 40 x ^ { 6 } y ^ { 4 } - 80 x ^ { 5 } y ^ { 5 } + 80 x ^ { 3 } y ^ { 6 } - 32 y ^ { 7 }$
B) $x ^ { 15 } - 10 x ^ { 12 } y ^ { 2 } + 40 x ^ { 9 } y ^ { 4 } - 80 x ^ { 6 } y ^ { 6 } + 80 x ^ { 3 } y ^ { 8 } - 32 y 10$
C) $x ^ { 15 } - 32 y ^ { 10 }$
D) $x ^ { 15 } - 2 x ^ { 12 } y ^ { 2 } + 4 x ^ { 9 } y ^ { 4 } - 8 x ^ { 6 } y ^ { 6 } + 16 x ^ { 3 } y ^ { 8 } - 32 y ^ { 10 }$

Question

Quizplus · Accepted Answer

The correct expansion using Pascal's triangle for the binomial $(x^3 - 2y^2)^5$ involves calculating each term based on the binomial theorem, which states that each term in the expansion is a combination of the form $C(n, k) \cdot a^{n-k} \cdot b^k$, where $n$ is the power of the binomial, $k$ is the term number, and $C(n, k)$ is the binomial coefficient from Pascal's triangle. For the 5th power, the coefficients are 1, 5, 10, 10, 5, 1. Applying these to the given binomial, while also considering the powers of $x^3$ and $-2y^2$, gives the correct expansion as in choice B.

Use Pascal's Triangle to Expand the Binomial (x3−2y2)5\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 5 }(x3−2y2)5

Use Pascal's Triangle to Expand the Binomial $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 5 }$