integer

Examples of integer in the following topics:

• Integer Coefficients and the Rational Zeros Theorem

  • Each solution to a polynomial, expressed as $x= \frac {p}{q}$, must satisfy that $p$ and $q$ are integer factors of $a_0$ and $a_n$, respectively.
  • If $a_0$ and $a_n$ are nonzero, then each rational solution $x= \frac {p}{q}$, where $p$ and $q$ are coprime integers (i.e. their greatest common divisor is $1$), satisfies:
  • Since any integer has only a finite number of divisors, the rational root theorem provides us with a finite number of candidates for rational roots.
  • When given a polynomial with integer coefficients, we can plug in all of these candidates and see whether they are a zero of the given polynomial.
  • Since every polynomial with rational coefficients can be multiplied with an integer to become a polynomial with integer coefficients and the same zeros, the Rational Root Test can also be applied for polynomials with rational coefficients.

• Zeroes of Polynomial Functions With Rational Coefficients

  • In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
  • Since q may be equal to 1, every integer is a rational number.
  • These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.
  • Zero divided by any other integer equals zero.
  • The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.

• Simplifying Exponential Expressions

  • Previously, we have applied these rules only to expressions involving integers.
  • However, they also apply to expressions involving a combination of both integers and variables.
  • In terms of conducting operations, exponential expressions that contain variables are treated just as though they are composed of integers.

• Simplifying Radical Expressions

  • Radical expressions containing variables can be simplified to a basic expression in a similar way to those involving only integers.
  • Radical expressions that contain variables are treated just as though they are integers when simplifying the expression.
  • This follows the same logic that we used above, when simplifying the radical expression with integers: $\sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$.

• Introduction to Factoring Polynomials

  • Factoring is the decomposition of an algebraic object, for example an integer or a polynomial, into a product of other objects, or factors, which when multiplied together give the original.
  • As an example, the integer $15$ factors as $3 \cdot 5$, and the polynomial $x^3 + 2x^2$ factors as $x^2(x+2)$.
  • In all cases, a product of simpler objects than the original (smaller integers, or polynomials of smaller degree) is obtained.
  • The aim of factoring is to reduce objects to "basic building blocks", such as integers to prime numbers, or polynomials to irreducible polynomials.

• Factors

  • Any whole number greater than one can be factored, which means it can be broken down into smaller integers.
  • Every positive integer greater than 1 has a distinct prime factorization.

• Fractions

  • A fraction represents a part of a whole and consists of an integer numerator and a non-zero integer denominator.
  • A common fraction, such as $\frac{1}{2}$, $ \frac{8}{5}$, or $\frac{3}{4}$, consists of an integer numerator (the top number) and a non-zero integer denominator (the bottom number).

• Total Number of Subsets

  • According to the theorem, it is possible to expand the power $(x + y)^n$ into a sum involving terms of the form $ax^by^c$, where the exponents $b$ and $c$ are nonnegative integers with $b+c=n$, and the coefficient $a$ of each term is a specific positive integer depending on $n$ and $b$.

• Sequences of Mathematical Statements

  • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2,4,6, \cdots )$.
  • So a sequence is formed by substituting integers $k$, $k + 1 $, $k + 2$ and so on into the mathematical statement.

• Sets of Numbers

  • For example: "$A$ is the set whose members are the first four positive integers."
  • For instance, the set of the first thousand positive integers may be specified extensionally as:
  • The set of integers includes all whole numbers (positive and negative), including $0$.
  • The set of integers is represented by the symbol $\mathbb{Z}$.