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 butthurt 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.
Also, to clarify the butthurt 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.
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.
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