What is the coefficient of x^9 in (2-x)^19?

Question

I got the answer: (2^10)*19!/(9!*10!) but the back of the book has a negative sign in front of the same answer....

I just wanted to know if my answer is correct or not?

Please help!

Update:

wait...so how did you guys get the (-1)^9...coz i got it as (-1)^10!!

Update 2:

oh....now i get it!! stupid me!! LOL! thanks for your help!!

Answer 1

Favorite Answer

Use binomial expansion. The two basic terms are 2 and -x.

The 10th term in the expension is C(19,9)2^(19-9) (-x)^9 = C(19,9)2^10 (-1)^9 x^9

So, the coefficient of x^9 is C(19,9)2^10 (-1)^9 = -(2^10)*19!/(9!*10!)

Answer 2

There's a negative in front of it because it is really (-x)^9, so the coefficient still keeps the negative since the power is an odd number; the negative doesn't cancel out. Other than that, yes your coefficient is correct.

C(19, 9) (2)^10 (-x)^9

The -1 is part of the x, since x is being subtracted, so it is really (2 + (-1x))^19. -1 is not part of 2, so it is not to the 10th power. -1x is one term, so the entire thing is placed to the 9th power. 2 is a different term, so only that is placed to the 10th power.

Answer 3

The book is correct. The power is x^9 which has a minus associated with it. The minus goes goes with the x and is raised to an odd power.

Answer 4

= 19C9 (2^10) (-x)^9

= -94595072x^9

Answer 5

It will have a negative sign.