Polynomial Nth Degree Equation

Overview

Engineering reference data for Polynomial Nth Degree Equation in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Polynomial Definitions by Degree

A polynomial function of degree $n$ is expressed as:

$f(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n$

where $n$ is a non-negative integer and $a_n \neq 0$ ensures the polynomial has exactly degree $n$.

Common special cases are defined by their degree:

• First Degree (Linear): $f(x) = a_0 + a_1 x$ with $a_1 \neq 0$. This represents a straight line.
• Second Degree (Quadratic): $f(x) = a_0 + a_1 x + a_2 x^2$ with $a_2 \neq 0$. This represents a parabola.
• Third Degree (Cubic): $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3$ with $a_3 \neq 0$. This can model curves with one or two inflection points.

The degree determines key properties, such as the maximum number of real roots (up to $n$) and the number of possible turning points (up to $n-1$).

References

• Original Source