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.

<p>Polynomial equations are used to encode a wide range of problems, from elementary word problems to complicated scientific problems.</p> Signup and view all the answers

<p>Polynomial functions are used in various settings, ranging from basic chemistry and physics to economics and social science.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>The two definitions of NP are equivalent.</p> Signup and view all the answers

<p>A nondeterministic Turing machine guesses a solution and then uses a deterministic algorithm to verify the solution.</p> Signup and view all the answers

<p>NP stands for 'nondeterministic, polynomial time', which reflects the two phases of the algorithm based on the Turing machine.</p> Signup and view all the answers

<p>NP consists of decision problems that can be solved or verified in polynomial time by a Turing machine.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>Fermat's right triangle theorem states that there is no solution in positive integers for $a^2 = b^4 + c^4$ and $a^4 = b^4 + c^2$.</p> Signup and view all the answers

<p>Fermat's Last Theorem states that $x^k + y^k = z^k$ is impossible in positive integers with $k &gt; 2$.</p> Signup and view all the answers

<p>The equation of a superellipse is $|x/a|^k + |y/b|^k = 1$.</p> Signup and view all the answers

<p>The squircle is a shape described by the equation $|x/a|^k + |y/b|^k = 1$.</p> Signup and view all the answers