Question

Find the term that is a constant in the binomial expansion of...

Answer 1

\[(x ^{3}-\frac{ 2 }{ x ^{2} })^{5}\]

Answer 2

What's your plan?

Answer 3

\[\left(\begin{matrix}5 \\ U\end{matrix}\right)(x ^{3})^{n-U}(-\frac{ 2 }{ x ^{2} })^{U}\]

Answer 4

Perfect. Now, were is \((x^{3})^{n-U}\cdot\left(\dfrac{1}{x^{2}}\right)^U = x^{0}\)

Answer 5

\[(-\frac{ x ^{2} }{ 2 })^{-U}\]?

Answer 6

nvm. I got it. Is -80 right?

Answer 7

We can worry about the constants later. We just need to know which termn it is for now.

\((x^{3})^{5}\cdot (x^{-2})^{0} = x^{15}\)
\((x^{3})^{4}\cdot (x^{-2})^{1} = x^{10}\)
\((x^{3})^{3}\cdot (x^{-2})^{2} = x^{5}\)
\((x^{3})^{2}\cdot (x^{-2})^{3} = x^{0}\)
\((x^{3})^{1}\cdot (x^{-2})^{4} = x^{-5}\)
\((x^{3})^{0}\cdot (x^{-2})^{5} = x^{-10}\)

Answer 8

But set 3(5-U)-2U to zero and solve for U. Then you can plug it back in and get -80 o-o

Answer 9

Fair enough. It's not perfectly clear to me what "find the term" means. Does it mean completely define it (-80) or just figure out which one it is (Term #4)?

Excellent work!

Answer 10

It means to find the term that is a constant, rather than a coefficient and a variable(s)

Answer 11

That doesn't clear it up at all.

If you find a rock under a flower, do you necessarily know what color the rock is? What if it's dark out?

It may seem a little silly, but exceptional clarity is a bit more difficult than allowing various assumptions. Let's see how it goes in this household.

Mom: Steve, will you help me find my car keys?
Steve: Sure. Where have you looked?
M: In the kitchen and the bedroom.
S: Okay, I'll go check the living room.
{a few moments pass}
S: Found them!
M: Where were they?
S: They're under the couch.
{a few moments pass}
M: Are you bringing the keys to me?
S: You said "find" them. I found them. They're still under the couch.

Maybe Steve was just being funny, but that doesn't mean Mom was being particularly clear.