Binomial Theorem: Finding Coefficient of Term

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dxoo

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Question Statement:
Find the coefficient of [MATH]x^6[/MATH] in the expansion of [MATH](2-x)(3x+1)^9[/MATH]"

My Approach:

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

So, the coefficient of [MATH]x^6[/MATH]:
[MATH]{9 \choose 3}(3^6)(x^6)+ {9 \choose 4}(3^5)(-x)[/MATH][MATH]= 61236x^6 - 30618x^6[/MATH][MATH]= 30618x^6[/MATH]
Therefore, according to my findings, the coefficient of [MATH]x^6[/MATH] is 30618.


However:
The provided answer in the textbook is [MATH]91854[/MATH]. Have I made a simple calculation error or is something fundamentally wrong in my approach. Any help is always appreciated, thanks!
 
dxoo said:
Question Statement:
Find the coefficient of [MATH]x^6[/MATH] in the expansion of [MATH](2-x)(3x+1)^9[/MATH]"

My Approach:

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

So, the coefficient of [MATH]x^6[/MATH]:
2*[MATH]{9 \choose 3}(3^6)(x^6)+ {9 \choose 4}(3^5)(-x)[/MATH][MATH]= 122472x^6 - 30618x^6[/MATH][MATH]= 91854x^6[/MATH]
Therefore, according to my findings, the coefficient of [MATH]x^6[/MATH] is 91854.


However:
The provided answer in the textbook is [MATH]91854[/MATH]. Have I made a simple calculation error or is something fundamentally wrong in my approach. Any help is always appreciated, thanks!
Click to expand...
You forgot to multiply by '2'

So, the coefficient of [MATH]x^6[/MATH]:
2*[MATH]{9 \choose 3}(3^6)(x^6)+ {9 \choose 4}(3^5)(-x)[/MATH][MATH]= 122472x^6 - 30618x^6[/MATH][MATH]= 91854x^6[/MATH]
Therefore, according to my findings, the coefficient of [MATH]x^6[/MATH] is 91854.
 
Last edited by a moderator:
dxoo said:
Silly mistake, thanks so much for your help!
Click to expand...
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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