polynomial

a mathematical term which has no use in real life,
unless u like finding out how much sp-ce something takes up.
which no one really does.
i like polynomials tho.they are easy to factor and stuff like dat.
“if a house is a rectangle 2x by 1.5x on a plot of land that was 23x by 21x, find out if they have enough land to plot a new garage, 17x by 11x.”
“who is they?!”
an expression such as
x^3 + 2x^2 – 6x – 2, is a cubic expression.
3x – 4 is linear
x^2 + 5x – 10 is quadratic

these types of expressions can be extended to include higher powers of x, such as
x^5 + 3x^3 – 2x^2 + 3
4x^6 + 2x^4 – x^3 + 5

expressions of this type are called polynomials.
note: polynomials do not have negative or fractional powers. n is a non-negative integer.

the degree of a polynomial is given by the highest power of the variable
an expression of the form
ax^n + bx^(n-1) + … + px^2 + qx + r
(where a, b, …, p, q, r are constants is called a polynomial in x)

3x^5 + 3x^4 – 2x^2 -1 is a polynomial of degree 5.

a constant expression is a polynomial of degree 0
a linear expression is a polynomial of degree 1
a quadratic expression is a polynomial of degree 2
a cubic expression is a polynomial of degree 3
etc.
a s-x position where botht the man and the woman contort their bodies in an exotic position resembling a polynomial
“oh bob, polynomial me harder”
” i polynomialed her hard last night……ooooooooo yeah!!!”
basically, the granddaddy of all equations across all spectrums of mathematics. if you can manipulate polynomials, you can represent almost all mathematical functions graphically.

also, to clarify the b-tthurt definition given ahead of me, polynomials actually are not useless in real life. and they are used for much more than the 6th grade application given. granted, said definer probably didn’t use them much in his career flipping burgers.

the form of a polynomial is as follows: given k is a positive integer, and c is a real number coefficient, and “a” is the value of x defined as the “center” of the polynomial…

a polynomial expression can be expressed by c(x-a)^k.

expression will be k+1 terms long, each k one integer less until k = 0.

example: k=3, a=0, c0=100,c1=1,c2=2, c3=3

3(x)^3 + 2(x)^2 + 1(x)^1 + 100(x)^0

all polynomials can be expressed in this manner. note that a cleaner way to express polynomials is through series, but i don’t know how to type that correctly.
fun fact 1) the idea of “imaginary” numbers stems from trying to factor polynomials of k>2.

it is a tedious process to find these imaginary roots, but to see this, you can plug in any cubic (k=3) to a graphing calculator. it will cross the x-axis twice, but, since it has a degree 3(k=3), it must have three roots. therefore, it will have two “real” roots, and one “imaginary” root. imaginary numbers have obscure use in high level electrical application.

fun fact 2) any function… any function (e^x, ln(x), sin(x)….) can be estimated using a polynomial function. the higher the degree(k), the more accurate the estimation will be.

this is can be done using whats called a “taylor approximation”.

it is really simple too, if you know what a derivative is, and how to take it. to get the taylor approximation, use the formula:

f^k(x) (take the kth derivative)
/
k!

this will give you c, and bam, you have your polynomial to whatever k you want.

if you want to show up your high school math teacher, you can use this to solve easier equations instead of dealing with stupid functions like arctan(x). highschool teachers are not required to take calculus ii, and that is where this simple formula is taught.