Question

The maximum number of regions formed by connecting (n) points of a circle can be obtained by the function $f\left(n\right)=\frac{1}{24}\left(n^4-6n^3+23n^2-18n+24\right)$. What is the degree of this polynomial.

Short answer

The degree of the polynomial is 4.

Step-by-step solution

Step 1. Write down the given information.

The given polynomial is $f\left(n\right)=\frac{1}{24}\left(n^4-6n^3+23n^2-18n+24\right)$.

Step 2. Calculation.

The given polynomial $f\left(n\right)=\frac{1}{24}\left(n^4-6n^3+23n^2-18n+24\right)$can be re-written as:

$\begin{array}{c}f\left(n\right)=\frac{1}{24}\left(n^4-6n^3+23n^2-18n+24\right)\\ f\left(n\right)=\frac{n^4}{24}-\frac{6n^3}{24}+\frac{23n^2}{24}-\frac{18n}{24}+\frac{24}{24}\\ f\left(n\right)=\frac{1}{24}{n}^4-\frac{1}{4}{n}^3+\frac{23}{24}{n}^2-\frac{3}{4}n+\mathrm{1....}\left(1\right)\end{array}$

From (1) it can be interpreted that the degree of polynomial is 4 whose leading coefficient is $\frac{1}{24}$.

Step 3. Conclusion.

The degree of the polynomial is $4$.