Polynomial

Table of Contents

Definition

an equation where terms are numbers multiplied by variable x. number * x it can be like $9x$ or $9x^2$ or $9x^0$

Requirements

the stipulations for what x are is:

• Leading Degree is not negative. It can be atleast x^0.

Degree

the degree is the highest power in the polynomial so for example, the polynomial $7x^4+6x^2+2$, the highest power is $4$ in $7x^4$ Orders are like degrees, but only for single terms/factors.

Leading Coefficient

The leading coefficient is the coefficient constant belonging to the variable with the highest Degree

Finding leading coefficient quickly in factor format

$(3x-4{)}^2\ast (5x^2-3)\ast (4x-8)$ we don’t care about the constants by them selves. so take them out $(3x{)}^2\ast (5x^2)\ast (4x)$ and then we can simplify now $9x^2\ast 5x^2\ast 4x=180x^5$ and so $5$ is the highest degree, the Leading Coefficient is by proxy $180$

Turning Points

A polynomial function will always have at max, (degree-1) number of turning points.

Global Minima And Maxima

Even degree polynomials always have ONE minima/maxima Odd degree polynomials always have ZERO minima/maxima, because domain and range are infinite.

Family

Polynomial functions that have the same roots

Even and Odd Functions

• Even and Odd Functions from Factored Form

Polynomial from finite differences

• Finite Differences
• Factorial
• Graphs In Different Degrees

Polynomial vs Non-polynomial

Polynomials have domains of x∈ℝ Non-polynomials have asymptotes