Rational Expressions - Revision Notes

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11.4Rational Expressions

In these revision notes for Rational Expressions, we cover the following key points:

  • What are rational expressions?
  • What do rational expressions have in common with rational numbers?
  • What are relatively prime factors?
  • How to convert an algebraic fraction into a rational expression
  • What the disallowed values are in a rational expression
  • How to simplify the rational expressions
  • How to multiply/divide rational expressions
  • How to add/subtract rational expressions
  • How to calculate the Least Common Multiple (LCM) of two or more monomials
  • What are complex fractions? How to deal with them?
  • How to calculate the value of recurring complex fractions

Rational Expressions Revision Notes

In algebra, a rational expression is a fraction that contains polynomials in both its numerator and denominator. A rational expression is different from an algebraic fraction in the sense that the variables in algebraic fractions can also have negative or fractional exponents (indices), unlike in rational expressions, where the exponents of the variables are positive integers or zero (in the constant term).

In short, a rational expression is an expression of the form P(x)/Q(x), where P(x) and Q(x) are both polynomials and Q(x) ≠ 0.

An algebraic fraction can sometimes turn into a rational expression through a series of operations. Obviously, not all algebraic fractions can become a rational expression, no matter how many transformations you make.

When the numerator and denominator of a rational expression have no common divisors (factors), we say they are relatively prime. In these cases no simplifications are possible. On the other hand, if the numerator and denominator have common factors, we can simplify the fraction by dividing both terms by any of the common factors. When dividing both terms with the greatest of the common factors (GCF), we obtain an equivalent fraction written in the simplest terms.

We must be careful to avoid the denominator being zero when dealing with rational expressions. Since the denominator in rational expression contains a polynomial, we have to reject the zeroes of the polynomial from the set of possible values of the variable.

By definition, the values of the variable for which the denominator of a rational expression becomes zero are called disallowed values. They represent the zeroes of the polynomial contained in the denominator of the given rational expression.

We use the following procedure to simplify a rational expression:

  1. Find the disallowed values first.
  2. Factorize completely both the numerator and denominator.
  3. Cancel out the common factors.
  4. Check whether the previous action affects the disallowed values of not.

Multiplication of rational expressions is carried out in the same way as the multiplication of rational numbers, where the numerators are multiplied separately as well as the denominators. In symbols, we have

R1 ∙ R2 = P1 (x)/Q1 (x)P2 (x)/Q2 (x) = P1 (x) ∙ P2 (x)/Q1 (x) ∙ Q2 (x)

From Arithmetic, it is known that division is the inverse operation of multiplication. In other words, division is the multiplication with the inverse. For the division of two numerical fractions, we have

a/b ÷ c/d = a/b × d/c = a × d/b × c

This rule is also valid for the division of rational expressions. Thus, replacing a, b, c and d in the above formula with P(x), Q(x), R(x) and S(x) respectively, yields

P(x)/Q(x) ÷ R(x)/S(x) = P(x)/Q(x)S(x)/R(x)

We relate the addition and subtraction of rational expressions with the addition and subtraction of fractions, where if the denominators are the same, we add or subtract only the numerators; otherwise, we have to find a common denominator before making any operation. Obviously, when finding the common denominator, it is necessary to multiply or divide both parts of one or more fractions with a suitable number so that to obtain equivalent fractions. In symbols, we have

x/z ± y/z = x ± y/z

for addition and subtraction of fractions with the same denominator and

w/x ± y/z = z ∙ w ± x ∙ y/x ∙ z

for addition and subtraction of fractions with different denominators.

The same procedure is also used when adding or subtracting rational expressions. The letters x, y, z and w are replaced with P(x), Q(x), etc., but the procedure is the same. Thus, we have

P(x)/Q(x) ± R(x)/Q(x) = P(x) ± R(x)/Q(x)

for addition and subtraction of rational expressions with the same denominator and

P(x)/Q(x) ± R(x)/S(x) = P(x) ∙ S(x) ± R(x) ∙ Q(x)/Q(x) ∙ S(x)

for addition and subtraction of rational expressions with different denominators.

The LCM of two algebraic terms is the smallest term that is divisible by all original terms. Here, we are interested only in monomials or polynomials, so the terms involved in the calculation of LCM belong to this category. The only difference is that when calculating the LCM of two monomials, we don't divide them by prime numbers but by coefficients and variables instead.

We use the LCM of two numbers, monomials or polynomials mainly to determine the least common denominator of two fractions or rational expressions when adding or subtracting them.

Complex fractions are those fractions that contain rational expressions in their numerator or/and denominator.

We use the fractions division rule to convert a complex fraction into a product of rational expressions. This means we have to invert down the bottom fraction and write the division of fractions represented by the central fraction bar into multiplication. In symbols, we write

a/b/c/d = a/bd/c = a ∙ d/b ∙ c

where a, b, c, d are all polynomials (they can also be monomials or simply integers, as they represent special cases of polynomials.

To avoid any possible confusion, it is better to write integers as fractions with denominator 1.

Sometimes you will encounter complex fractions that follow the same pattern up to infinity. Although at the first sight they look as impossible to handle with because of they recurrence that extends up to infinity, it is possible to calculate their value following the procedure below.

Step 1: Express the original complex fraction as a variable. For example, denote the original fraction as x. In our example, we have

Step 2: Express the next inner sequence by the same letter. This is because the complex fraction extends to infinity, so infinity minus one and infinity itself are the same thing. In our case, we have

Step 3: Solve the equation obtained. The value of the variable represents the value of the recurring complex fraction.

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