Find the coefficient of $x^{70}$ in the product $(x-1)(x^2-2)(x^3-3)…(x^{12}-12)$.
The above equation is a polynomial of degree 78 . But I am not able to find the coefficient of $x^{70}$
Asked
Active
Viewed 156 times
2
-
2So total polynomial degree is 78, and you want $x^{70}$. It means replacing some $x^i$ with their corresponding integer ($i$), where $\sum i=8$. Can you take it from there? – Gareth Ma Apr 14 '20 at 14:36
1 Answers
5
So total polynomial degree is 78, and you want $x^{70}$. It means replacing some $x^i$ with their corresponding integer ($-i$), where $\sum i=8$. Noticing that $1+2+3+4=10>8$, there are very few cases, and we can list them.
When I say replace, I mean replacing the term in the product $x^1x^2\cdots x^{12}$.
- When we replace 1 term:
We must replace $x^8$ by -8, and we get the term $\fbox{$-8x^{70}$}$ from this.
- When we replace 2 terms:
We can replace $x^ix^{8-i}$ by i(8-i), where $i$ can range from $1$ to $3$. We get the term $(1\cdot7+2\cdot6+3\cdot5)x^{70}=\fbox{$34x^{70}$}$ from this.
- When we replace 3 terms:
Scary? No! The only (unordered) triples that sum up to $8$ are $(1, 2, 5)$ and $(1, 3, 4)$! These contribute to the term $((-1)(-2)(-5)+(-1)(-3)(-4))x^{70}=\fbox{$-22x^{70}$}$.
To conclude, we get that $4$ is the coefficient of $x^{70}$.
Gareth Ma
- 1,853