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Zeroes of Polynomial Functions With Rational Coefficients
Polynomials with rational coefficients should be treated and worked the same as other polynomials.
Learning Objective

Extend the techniques of finding zeros to polynomials with rational coefficients
Key Points
 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.
 A real number that is not rational is called irrational. Irrational numbers include √2, π, and e.
 Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations.
Terms

quotient
The number resulting from the division of one number or expression by another.

irrational number
Any real number that cannot be expressed as a ratio of two integers.
Full Text
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. The set of all rational numbers is usually denoted by a boldface Q (or Unicode ℚ). It was thus named in 1895 by Peano after quoziente, Italian for "quotient".
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.
A real number that is not rational is called irrational. Irrational numbers include √2, π, and e. The decimal expansion of an irrational number continues forever without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.
Zero divided by any other integer equals zero. Therefore zero is a rational number, but division by zero is undefined.
The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, the adjective rational often means that the underlying field considered is the field Q of rational numbers. Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals". However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.
Finding Zeroes of a Polynomial with Rational Coefficients
Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations. Multiplying fractions a/b times c/d gives (ac)/(bd), whereas if one wanted to add a/b plus c/d, first convert them into ad/bd and cb/db, giving (ad+cb)/(db).
For example, the polynomial
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Key Term Reference
 base
 Appears in these related concepts: Changing Logarithmic Bases, Logarithms of Powers, and Rational Exponents
 coefficient
 Appears in these related concepts: Factoring General Quadratics, What Are Polynomials?, and Introduction to Variables
 denominator
 Appears in these related concepts: Complex Conjugates and Division, Rational Equations, and Fractions Involving Radicals
 e
 Appears in these related concepts: Natural Logarithms, The Number e, and Business Stakeholders: Internal and External
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 finite
 Appears in these related concepts: The Sample Average, Introduction to Sequences, and Summing Terms in an Arithmetic Sequence
 fraction
 Appears in these related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 function
 Appears in these related concepts: Solving Differential Equations, Visualizing Domain and Range, and The Vertical Line Test
 integer
 Appears in these related concepts: Scientific Notation, Finding a Specific Term, and Binomial Expansions and Pascal's Triangle
 operation
 Appears in these related concepts: Designing the Operation, Introduction to Exponents, and Job security and people
 polynomial
 Appears in these related concepts: Domains of Rational and Radical Functions, The Remainder Theorem and Synthetic Division, and Finding Polynomials with Given Zeroes
 rational function
 Appears in these related concepts: Introduction to Rational Functions, Asymptotes, and Solving Problems with Rational Functions
 rational number
 Appears in these related concepts: The Intermediate Value Theorem, Radical Functions, and Basic Operations
 real number
 Appears in these related concepts: Zeroes of Polynomial Functions with Real Coefficients, Solving Problems with Inequalities, and Introduction to Complex Numbers
 real numbers
 Appears in these related concepts: Introduction to Domain and Range, Piecewise Functions, and Linear Inequalities
 sequence
 Appears in these related concepts: Series, Summing an Infinite Series, and The General Term of a Sequence
 set
 Appears in these related concepts: Sequences of Mathematical Statements, Sequences, and Sets of Numbers
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
 zero
 Appears in these related concepts: Rational Inequalities, The Discriminant, and Other Equations in Quadratic Form
Sources
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Source: Boundless. “Zeroes of Polynomial Functions With Rational Coefficients.” BOOKS. Boundless, 02 Jun. 2016. Retrieved 30 Jul. 2016 from https://www.boundless.com/users/317520/textbooks/books/polynomialandrationalfunctions7/zeroesofpolynomialfunctionsandtheirtheorems48/zeroesofpolynomialfunctionswithrationalcoefficients1495520/