I just started learning about using factorial form of the binomial theorem to expand, but my textbook didn't really explain everything well so I'm a bit confused. I was told to find the eighth term of (3a+b)^9. Can someone check my answer? Thanks.
9!
----- 3a^2 (b)^7
(9-7)7!
9 × 8 × 7
---------------- 3a^2 (b)^7
2 × 7
72
------- 3a^2 (b)^7
2
216
-------- a^2 (b)^7
2
108a^2 b^7
9!/(7!2!) = (9*8)/(1*2) = 36 = 9C2
So, the 8th term is
36 (3a)^2 b^7 = 36*9 a^2 b^7 = 324 a^2 b^7
Note that (3a)^2 ≠ 3a^2
it is (3a)(3a) = 9a^2
Thank you so much for your help. I now understand how to do this!!!!
To find the eighth term of (3a+b)^9, you can use the factorial form of the binomial theorem. The formula for the term in position k is given by:
C(n,k) * (a^(n-k)) * (b^k)
Where n is the exponent of the binomial and k is the position of the term (starting from 0).
In this case, n = 9 and k = 8. Plugging these values into the formula, we get:
C(9,8) * (3a)^(9-8) * (b^8)
Now let's calculate each part:
C(9,8) = 9! / (8! * (9-8)!) = 9
(3a)^(9-8) = (3a)^1 = 3a
(b^8) = b^8
Now we can substitute these values back into the formula:
9 * 3a * b^8
Simplifying, we get:
27a * b^8
Therefore, the correct answer is 27a * b^8, and not 108a^2 * b^7 as you have calculated.