# 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).