Question

What is the second term of (2x 2 ‑ 3y)^6 ? Your answer should be in the form (sign)(numerical factors)(actual powers of variables).

Answer 1

`-6(2^5)(3)x^(10)y`.

Explanation:

Use the binomial theorem: given any variables a,b and a positive integer n,
`(a+b)^n=\sum_(k=0)^n \binom{n}{k}a^(n-k)b^(k)`.

In this case, take
`a=2x^2,b=-3y,n=6`

Usually, the terms on a polynomial of two variables are ordered starting with the highest power of x (x^12 in this case) and the lowest power of y. The powers of x decrease and the powers of y increase, so the last term has the highest power of y and the lowest power of x .

Then, the second term is k=1 as it has the second highest power of x, so replacing these values this term is:

`\binom{6}{1}(2x^2)^5 (-3y)^(1)=6(2^5 x^(10))(-3y)=-6(2^5)(3)x^(10)y`.