Mastering Polynomials

A polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial with a single indeterminate is $x^2 - 4x + 7$.

An example of a polynomial with three indeterminates is $x^3 + 2xyz^2 - yz + 1$.

Polynomials appear in many areas of mathematics and science, such as forming polynomial equations, defining polynomial functions, calculus, numerical analysis, constructing polynomial rings, and algebraic varieties.

Polynomial equations are used to encode a wide range of problems, from elementary word problems to complicated scientific problems.

Polynomial functions are used in various settings, ranging from basic chemistry and physics to economics and social science.

NP is the set of decision problems for which the problem instances, where the answer is 'yes', have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.

The two definitions of NP are equivalent.

A nondeterministic Turing machine guesses a solution and then uses a deterministic algorithm to verify the solution.

NP stands for 'nondeterministic, polynomial time', which reflects the two phases of the algorithm based on the Turing machine.

NP consists of decision problems that can be solved or verified in polynomial time by a Turing machine.

Faulhaber's formula expresses $1^k + 2^k + 3^k + \cdots + n^k$ as a polynomial in $n$, or alternatively in terms of a Bernoulli polynomial.

Fermat's right triangle theorem states that there is no solution in positive integers for $a^2 = b^4 + c^4$ and $a^4 = b^4 + c^2$.

Fermat's Last Theorem states that $x^k + y^k = z^k$ is impossible in positive integers with $k > 2$.

The equation of a superellipse is $|x/a|^k + |y/b|^k = 1$.

The squircle is a shape described by the equation $|x/a|^k + |y/b|^k = 1$.