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The multiplicative identity of R[x] is the polynomial x 0; that is, x 0 times any polynomial p(x) is just p(x). [2] Also, polynomials can be evaluated by specializing x to a real number. More precisely, for any given real number r, there is a unique unital R-algebra homomorphism ev r : R[x] → R such that ev r (x) = r. Because ev r is unital ...
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab
For example, using single-precision IEEE arithmetic, if x = −2 −149, then x/2 underflows to −0, and dividing 1 by this result produces 1/(x/2) = −∞. The exact result −2 150 is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.
Graphs of y = b x for various bases b: base 10, base e, base 2, base 1 / 2 . Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
Because (a + 1) 2 = a, a + 1 is the unique solution of the quadratic equation x 2 + a = 0. On the other hand, the polynomial x 2 + ax + 1 is irreducible over F 4, but it splits over F 16, where it has the two roots ab and ab + a, where b is a root of x 2 + x + a in F 16. This is a special case of Artin–Schreier theory.
[2] A root of a polynomial is a zero of the corresponding polynomial function . [ 1 ] The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree , and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an ...
Expanding (x + y) n yields the sum of the 2 n products of the form e 1 e 2... e n where each e i is x or y. Rearranging factors shows that each product equals x n−k y k for some k between 0 and n. For a given k, the following are proved equal in succession: the number of terms equal to x n−k y k in the expansion
In a totally ordered ring, x 2 ≥ 0 for any x. Moreover, x 2 = 0 if and only if x = 0. In a supercommutative algebra where 2 is invertible, the square of any odd element equals zero. If A is a commutative semigroup, then one has , = = =.