Question

Binomial Theorem help!
Show the work to find the third term of
(2x-3y)^7. Then use your calculator to help find the term.

Answer 1

Lol I've seen this exact question posted several times in the past few days :)
I wonder if you're all in the same class or something hehe.

Answer 2

Ok so Binomial Theorem:

Answer 3

\[\large\rm (\color{orangered}{a}+\color{royalblue}{b})^n\quad=\sum_{k=0}^n \left(\begin{matrix}n \\ \rm k\end{matrix}\right)\left(\color{orangered}{a}\right)^{n-k}\left(\color{royalblue}{b}\right)^k\]

Answer 4

So for our specific problem we have:\[\large\rm (\color{orangered}{2x}+\color{royalblue}{-3y})^7\quad=\sum_{k=0}^7 \left(\begin{matrix}7 \\ \rm k\end{matrix}\right)\left(\color{orangered}{2x}\right)^{7-k}\left(\color{royalblue}{-3y}\right)^k\]

Answer 5

If you ignore the summation, then this gives us the expression for our `general term`.\[\large\rm general~term:\qquad \left(\begin{matrix}7 \\ \rm k\end{matrix}\right)(2x)^{7-k}(-3y)^k\]

Answer 6

Recall that k starts counting from 0, not from 1!
So our first term is the 0th term,
our second term being the k=1 term,
and our third term being the k=2 term.

So our third term will correspond to k=2, not k=3!

Answer 7

So plug k=2 into your general term, and simplify.