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
Interactive Graph: Multiplying Fractions
Graph of a polynomial with the quadratic equation of
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: Introduction to Variables, Factoring General Quadratics, and Balancing Chemical Equations
 denominator
 Appears in these related concepts: Rational Equations, Complex Conjugates and Division, and Fractions Involving Radicals
 e
 Appears in these related concepts: The Number e, Natural Logarithms, and Business Stakeholders: Internal and External
 equation
 Appears in these related concepts: Graphs of Equations as Graphs of Solutions, Equations and Inequalities, 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: Functions and Their Notation, The Vertical Line Test, and What is a Linear Function?
 integer
 Appears in these related concepts: Binomial Expansions and Pascal's Triangle, Scientific Notation, and Finding a Specific Term
 polynomial
 Appears in these related concepts: Finding Polynomials with Given Zeros, Domains of Rational and Radical Functions, and The Remainder Theorem and Synthetic Division
 rational function
 Appears in these related concepts: Asymptotes, Solving Problems with Rational Functions, and Introduction to Rational Functions
 rational number
 Appears in these related concepts: Radical Functions, The Intermediate Value Theorem, and Basic Operations
 real number
 Appears in these related concepts: Introduction to Complex Numbers, Solving Problems with Inequalities, and Zeros of Polynomial Functions with Real Coefficients
 real numbers
 Appears in these related concepts: Linear Inequalities, Piecewise Functions, and Introduction to Domain and Range
 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: The 22nd Amendment, Basics of Graphing Polynomial Functions, and Democracy
 zero
 Appears in these related concepts: Rational Inequalities, Historical Traditions of Numerical Systems, and Zeroes of Linear Functions
 zeros
 Appears in these related concepts: Applications of the Parabola, Parts of a Parabola, and A Graphical Interpretation of Quadratic Solutions
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