Binomial Theorem: Finding Coefficient of Term

dxoo

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Jul 16, 2020
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Question Statement:
Find the coefficient of \(\displaystyle x^6\) in the expansion of \(\displaystyle (2-x)(3x+1)^9\)"

My Approach:

\(\displaystyle (3x+1)^9 = {9 \choose 0}(3x)^9 + {9 \choose 1}(3x)^8 + {9 \choose 2}(3x)^7 + {9 \choose 3}(3x)^6 + {9 \choose 4}(3x)^5 + ....\)

And the only numbers that could have the variable of \(\displaystyle x^6\) is the fourth term \(\displaystyle {9 \choose 3}(3x)^6\)
or the fifth term \(\displaystyle {9 \choose 4}(3x)^5\) multiplied by the \(\displaystyle x\) in the binomial \(\displaystyle (2-x)\).

So, the coefficient of \(\displaystyle x^6\):
\(\displaystyle {9 \choose 3}(3^6)(x^6)+ {9 \choose 4}(3^5)(-x)\)
\(\displaystyle = 61236x^6 - 30618x^6\)

\(\displaystyle = 30618x^6\)

Therefore, according to my findings, the coefficient of \(\displaystyle x^6\) is 30618.



However:
The provided answer in the textbook is \(\displaystyle 91854\). Have I made a simple calculation error or is something fundamentally wrong in my approach. Any help is always appreciated, thanks!
 

Subhotosh Khan

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22,083
Question Statement:
Find the coefficient of \(\displaystyle x^6\) in the expansion of \(\displaystyle (2-x)(3x+1)^9\)"

My Approach:

\(\displaystyle (3x+1)^9 = {9 \choose 0}(3x)^9 + {9 \choose 1}(3x)^8 + {9 \choose 2}(3x)^7 + {9 \choose 3}(3x)^6 + {9 \choose 4}(3x)^5 + ....\)

And the only numbers that could have the variable of \(\displaystyle x^6\) is the fourth term \(\displaystyle {9 \choose 3}(3x)^6\)
or the fifth term \(\displaystyle {9 \choose 4}(3x)^5\) multiplied by the \(\displaystyle x\) in the binomial \(\displaystyle (2-x)\).

So, the coefficient of \(\displaystyle x^6\):
2*\(\displaystyle {9 \choose 3}(3^6)(x^6)+ {9 \choose 4}(3^5)(-x)\)
\(\displaystyle = 122472x^6 - 30618x^6\)

\(\displaystyle = 91854x^6\)

Therefore, according to my findings, the coefficient of \(\displaystyle x^6\) is 91854.



However:
The provided answer in the textbook is \(\displaystyle 91854\). Have I made a simple calculation error or is something fundamentally wrong in my approach. Any help is always appreciated, thanks!
You forgot to multiply by '2'

So, the coefficient of \(\displaystyle x^6\):
2*\(\displaystyle {9 \choose 3}(3^6)(x^6)+ {9 \choose 4}(3^5)(-x)\)
\(\displaystyle = 122472x^6 - 30618x^6\)
\(\displaystyle = 91854x^6\)

Therefore, according to my findings, the coefficient of \(\displaystyle x^6\) is 91854.
 
Last edited:

dxoo

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Joined
Jul 16, 2020
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Silly mistake, thanks so much for your help!
 

Subhotosh Khan

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Silly mistake, thanks so much for your help!
Yes... but you should have caught it by observing - by how much your answer is different from the given one (that's how I caught it).
 
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