# Rational Coefficients

## Polynomials with rational coefficients should be treated and worked the same as other polynomials.

#### 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

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

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

#### Figures

1. ##### Interactive Graph: Multiplying Fractions

Graph of a polynomial with the quadratic equation of $y=\frac{2x^2}{9}+\frac{7x}{3}+6$. We can graph this equation, and in doing so see where it intercepts the y axis, as a means of checking our solutions to this problem.

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 $\frac {2x^2}9+\frac {7x}3+6$ can be factored to give $(\frac x3 +2)(\frac {2x}3+3)$. By setting each term to zero, it can be found that the zeros for this equation are x=-6 and x=-9/2. This matches what is observed graphically, as shown in Figure 1.

#### Key Term Glossary

algebraic
or function}} Containing only numbers, letters and arithmetic operators.
##### Appears in these related concepts:
base
A number raised to the power of an exponent.
##### Appears in these related concepts:
coefficient
a constant by which an algebraic term is multiplied.
##### Appears in these related concepts:
denominator
The number or expression written below the line in a fraction (thus 2 in ½).
##### Appears in these related concepts:
e
The base of the natural logarithm, 2.718281828459045…
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
##### Appears in these related concepts:
finite
Limited, constrained by bounds, impermanent.
##### Appears in these related concepts:
fraction
A ratio of two numbers, the numerator and the denominator, usually written one above the other and separated by a horizontal bar.<!--rational number (a rational number can always be expressed as a fraction using integers, but some fractions can even contain irrational numbers, (eg: pi/4, sqrt(2)/2; Also: ⅔/3 )-->
##### Appears in these related concepts:
function
A relation in which each element of the input is associated with exactly one element of the output.
##### Appears in these related concepts:
integer
An element of the infinite and numerable set {...,-3,-2,-1,0,1,2,3,...}.
##### Appears in these related concepts:
irrational number
Any real number that cannot be expressed as a ratio of two integers.
##### Appears in these related concepts:
operation
A procedure for generating a value from one or more other values.
##### Appears in these related concepts:
polynomial
an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as $a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0$. Importantly, because all exponents are positive, it is impossible to divide by x.
##### Appears in these related concepts:
quotient
The number resulting from the division of one number or expression by another.
##### Appears in these related concepts:
rational function
Any function whose value can be expressed as the quotient of two polynomials (except division by zero).
##### Appears in these related concepts:
rational number
A real number that can be expressed as the ratio of two integers.
##### Appears in these related concepts:
real number
An element of the set of real numbers.  The set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers.
##### Appears in these related concepts:
real numbers
The smallest set containing all limits of convergent sequences of rational numbers.
##### Appears in these related concepts:
sequence
A set of things next to each other in a set order; a series
##### Appears in these related concepts:
set
A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects that may be contained within it.
##### Appears in these related concepts:
term
any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.
##### Appears in these related concepts:
zero
Also known as a root, a zero is an x value at which the function of x is equal to 0.