Yup, it works for ##n=2## and ##n=3##, so I'd say it could be generalized to any odd and even integer index...fresh_42 said:Simply try some low numbers for ##n##. And of course, it is allow in case ##a=b##.
So, for example it would be true that -(2-1)*2 = (1-2)*2, yes?greg_rack said:Yup, it works for ##n=2##
Nevermind, I wrote without thinking...fresh_42 said:How can it work for ##n=2## if squares are non negative, but the left hand side is?
Yes, you have ## -(a-b)^n=(-1)^n(a-b)^n##. Now comes the important step which is usually forgotten and a never ending well of mistakes!greg_rack said:Nevermind, I wrote without thinking...
By the way:
$$(b-a)^n=[-1(a-b)]^n=(-1)^n(a-b)^n$$
which is equal to ##-(a-b)^n## only for odd indexes.
Got it, thanks a lot!fresh_42 said:Yes, you have ## -(a-b)^n=(-1)^n(a-b)^n##. Now comes the important step which is usually forgotten and a never ending well of mistakes!
Case 1: ##(a-b)^n = 0 \Longrightarrow a=b##
Case 2: ##(a-b)^n\neq 0 \Longrightarrow (-1)^n=-1 \Longrightarrow n \text{ odd }##
greg_rack said:Yup, it works for n = 2 and n = 3
A polynomial identity is a mathematical statement that equates two polynomial expressions. It is similar to an equation, but instead of solving for a specific value, it shows that the two expressions are always equal, regardless of the values of the variables.
A polynomial identity is different from a polynomial equation in that it is always true, while a polynomial equation may only be true for certain values of the variables. Additionally, a polynomial identity is often used to prove other mathematical theorems, while a polynomial equation is used to solve for a specific value.
One example of a polynomial identity is the Binomial Theorem, which states that (x + y)^n = Σ(n choose k) * x^(n-k) * y^k, where n is a positive integer and (n choose k) represents the binomial coefficient. This identity can be used to expand binomial expressions, such as (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3.
Polynomial identities are used in various fields of science and engineering, such as physics, chemistry, and computer science. They are often used to simplify complex mathematical expressions and to prove important theorems. For example, in physics, polynomial identities are used to simplify equations of motion and to derive formulas for physical laws.
Yes, there are many practical applications of polynomial identities. In engineering, they are used to design and analyze circuits, control systems, and other complex systems. In computer science, they are used in data compression, error correction, and cryptography. In economics, polynomial identities are used to model and analyze financial data. Overall, polynomial identities play a crucial role in many real-world applications.