Binomial Theorem: Finding Coefficient of Term

dxoo

New member
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

Super Moderator
Staff member
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

New member
Silly mistake, thanks so much for your help!

Subhotosh Khan

Super Moderator
Staff member
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).