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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: Temple Architecture in the Greek Orientalizing Period, Rules for Exponent Arithmetic, and Rational Exponents
 coefficient
 Appears in these related concepts: Factoring General Quadratics, Introduction to Variables, and Balancing Chemical Equations
 denominator
 Appears in these related concepts: Rational Equations, Fractions Involving Radicals, and Complex Conjugates and Division
 e
 Appears in these related concepts: Natural Logarithms, Business Stakeholders: Internal and External, and The Number e
 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: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 integer
 Appears in these related concepts: Scientific Notation, Binomial Expansions and Pascal's Triangle, and Finding a Specific Term
 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, Simplifying, Multiplying, and Dividing, and Partial Fractions
 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, Basic Operations, and Radical Functions
 real number
 Appears in these related concepts: Solving Problems with Inequalities, Introduction to Complex Numbers, and Zeroes of Polynomial Functions with Real Coefficients
 real numbers
 Appears in these related concepts: Piecewise Functions, Introduction to Domain and Range, 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, Sets of Numbers, and Sequences of Mathematical Statements
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
 zero
 Appears in these related concepts: Rational Inequalities, Historical Traditions of Numerical Systems, and Other Equations in Quadratic Form
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Source: Boundless. “Zeroes of Polynomial Functions With Rational Coefficients.” BOOKS. Boundless, 08 Aug. 2016. Retrieved 22 Oct. 2016 from https://www.boundless.com/users/317520/textbooks/books/polynomialandrationalfunctions7/zeroesofpolynomialfunctionsandtheirtheorems48/zeroesofpolynomialfunctionswithrationalcoefficients1495520/