Watch
Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made.
Favorite
Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students.
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
Key Term Reference
 base
 Appears in these related concepts: Strong Bases, The Role of the Kidneys in AcidBase Balance, and Biology: DNA Structure and Replication
 coefficient
 Appears in these related concepts: Standard Form and Completing the Square, Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1, and Balancing Chemical Equations
 denominator
 Appears in these related concepts: Low Voter Turnout, Complex Conjugates and Division, and Rational Equations
 e
 Appears in these related concepts: Derivatives of Exponential Functions, Natural Logarithms, and e
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 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: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 integer
 Appears in these related concepts: Exact Numbers, Scientific Notation, and Finding a Specific Term
 operation
 Appears in these related concepts: Outsourcing, Designing the Operation, and Integer Exponents
 polynomial
 Appears in these related concepts: Basics of Graphing Polynomial Functions, Polynomials: Introduction, Addition, and Subtraction, and Greatest Common Factor and Factoring by Grouping
 rational function
 Appears in these related concepts: Finding the Domain of a Rational Function, Asymptotes, and Solving Problems with Rational Functions
 rational number
 Appears in these related concepts: The Intermediate Value Theorem, Rational Exponents, and Real Numbers: Basic Operations
 real number
 Appears in these related concepts: Graphing the Normal Distribution, Solving Problems with Inequalities, and The ComplexNumber System
 real numbers
 Appears in these related concepts: Finding Domains of Functions, Zeroes of Polynomial Functions with Real Coefficients, and Linear Inequalities
 sequence
 Appears in these related concepts: Series, Summing an Infinite Series, and Finding the General Term
 set
 Appears in these related concepts: Sequences of Statements, Expressions and Sets of Numbers, and Executive Function and Control
 term
 Appears in these related concepts: Arithmetic Sequences, The Executive Departments, and The 22nd Amendment
 zero
 Appears in these related concepts: Finding Polynomials with Given Zeroes, The Discriminant, and Reducing Equations to a Quadratic
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources:
Cite This Source
Source: Boundless. “Rational Coefficients.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 30 Nov. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/polynomialandrationalfunctions4/zeroesofpolynomialfunctionsandtheirtheorems31/rationalcoefficients1545520/