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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

irrational number
Any real number that cannot be expressed as a ratio of two integers.

quotient
The number resulting from the division of one number or expression by another.
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 this related concepts: Integer Exponents, Logarithms of Powers, and Balancing Redox Equations
 coefficient
 Appears in this related concepts: Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1, Balancing Chemical Equations, and Polynomials: Introduction, Addition, and Subtraction
 denominator
 Appears in this related concepts: Complex Conjugates and Division, Rational Equations, and Rationalizing Denominators or Numerators
 e
 Appears in this related concepts: Derivatives of Exponential Functions, Derivatives of Logarithmic Functions, and Natural Logarithms
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 finite
 Appears in this related concepts: The Sample Average, Introduction to Sequences, and Summing Terms in an Arithmetic Sequence
 fraction
 Appears in this related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 function
 Appears in this related concepts: Solving Differential Equations, Four Ways to Represent a Function, and Average Value of a Function
 integer
 Appears in this related concepts: Scientific Notation, Total Number of Subsets, and Finding a Specific Term
 operation
 Appears in this related concepts: Designing the Operation, The Importance of Clarity in Professional Settings, and Converting between Exponential and Logarithmic Equations
 polynomial
 Appears in this related concepts: Differentiation Rules, Partial Fractions, and Greatest Common Factor and Factoring by Grouping
 rational function
 Appears in this related concepts: Finding the Domain of a Rational Function, Asymptotes, and Solving Problems with Rational Functions
 rational number
 Appears in this related concepts: The Intermediate Value Theorem, Rational Exponents, and Real Numbers: Basic Operations
 real number
 Appears in this related concepts: Graphing the Normal Distribution, Solving Problems with Inequalities, and The ComplexNumber System
 real numbers
 Appears in this related concepts: Piecewise Functions, Linear Inequalities, and Factoring Trinomials of the Form ax^2 + bx + c; Perfect Squares
 sequence
 Appears in this related concepts: Series, Summing an Infinite Series, and Finding the General Term
 set
 Appears in this related concepts: Sequences of Statements, Sequences, and Expressions and Sets of Numbers
 term
 Appears in this related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
 zero
 Appears in this related concepts: Zeroes of Linear Functions, Rational Inequalities, and Finding Zeroes of Factored Polynomials
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Cite This Source
Source: Boundless. “Rational Coefficients.” Boundless Algebra. Boundless, 03 Jul. 2014. Retrieved 20 May. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/polynomialandrationalfunctions4/zeroesofpolynomialfunctionsandtheirtheorems31/rationalcoefficients1545520/