rational expression

Examples of rational expression in the following topics:

• Simplifying, Multiplying, and Dividing Rational Expressions

  • As a first example, consider the rational expression $\frac { 3x^3 }{ x }$.
  • Rational expressions can be multiplied and divided in a similar manner to fractions.
  • We follow the same rules to multiply two rational expressions together.
  • Dividing rational expressions follows the same rules as dividing fractions.
  • The same applies to dividing rational expressions; the first expression is multiplied by the reciprocal of the second.

• Rational Algebraic Expressions

  • The addition and subtraction of rational expressions are bound by all of the same rules as the addition and subtraction of fractions.
  • The key is finding the least common denominator of the two rational expressions: the smallest multiple of both denominators.
  • When applying this strategy to rational expressions, first look at the denominators of the two rational expressions and see if they are the same.
  • Rather, we will be looking for monomial and binomial factors that are common to both rational expressions.
  • The rational expressions therefore become:

• Rational Inequalities

  • Because a rational expression consists of the ratio of two polynomials, the zeroes for both polynomials will be needed.
  • The product of a positive and three negatives is negative, so the rational expression becomes negative as it crosses $x=-3$ in the rightward direction.
  • The same process can be used to determine that the rational expression is positive after passing the zero at $x=-2$, is negative after passing $x=1$, and is positive after passing $x=2$.
  • Thus we can conclude that for $x$ values on the open interval from $-\infty$ to $-3$, the rational expression is negative.
  • Solve for the zeros of a rational inequality to find its solution

• Rational Equations

  • A rational equation sets two rational expressions equal to each other and involves unknown values that make the equation true.
  • For an equation that involves two fractions or rational expressions, cross-multiplying is a helpful strategy for simplifying the equation or determining the value of a variable.
  • Notice that the rational expressions on both sides of the equal sign have the same denominator.
  • If you have a rational equation where the denominators on either side of the equation are the same, then their respective numerators must also be the same value, even though they might be expressed in different terms.

• Solving Equations with Rational Expressions; Problems Involving Proportions

  • Rational expressions, like proportions, are extremely useful applications of algebra, that can be solved using simple algebraic techniques.
  • A rational equation means that you are setting two rational expressions equal to each other.
  • This leads us to a very general rule: If you have a rational equation where the denominators are the same, then the numerators must be the same.
  • For example, consider the rational equation $\displaystyle \frac 3 {x^2+12x+36} = \frac {4x}{x^3+4x^2-12x}$
  • Solve equations with rational expressions (proportions) by finding the LCD or by cross-multipliation

• Domains of Rational and Radical Functions

  • Rational and radical expressions have restrictions on their domains which can be found algebraically or graphically.
  • A rational expression is one which can be written as the ratio of two polynomial functions.
  • Despite being called a rational expression, neither the coefficients of the polynomials nor the values taken by the function are necessarily rational numbers.
  • In the case of one variable, $x$, an expression is called rational if and only if it can be written in the form:
  • The domain of a rational expression of is the set of all points for which the denominator is not zero.

• Rational Exponents

  • Rational exponents are another method for writing radicals and can be used to simplify expressions involving both exponents and roots.
  • A rational exponent is a rational number that provides another method for writing roots.
  • The following are rules for operations on numbers with rational exponents.
  • This expression can be rewritten using the rule for dividing numbers with rational exponents:
  • Relate rational exponents to radicals and the rules for manipulating them

• Zeroes of Polynomial Functions With Rational Coefficients

  • 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.
  • 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.

• Integer Coefficients and the Rational Zeros Theorem

  • Each solution to a polynomial, expressed as $x= \frac {p}{q}$, must satisfy that $p$ and $q$ are integer factors of $a_0$ and $a_n$, respectively.
  • In algebra, the Rational Zero Theorem, or Rational Root Theorem, or Rational Root Test, states a constraint on rational solutions (also known as zeros, or roots) of the polynomial equation
  • Since any integer has only a finite number of divisors, the rational root theorem provides us with a finite number of candidates for rational roots.
  • We can use the Rational Root Test to see whether this root is rational.
  • Use the Rational Zeros Theorem to find all possible rational roots of a polynomial

• Fractions Involving Radicals

  • Root rationalization is a process by which any roots in the denominator of an irrational fraction are eliminated.
  • When we are given expressions that involve radicals in the denominator, it makes it easier to evaluate the expression if we rewrite it in a way that the radical is no longer in the denominator.
  • This process is called rationalizing the denominator.
  • Let's look at an example to illustrate the process of rationalizing the denominator.
  • Therefore, multiply the top and bottom of the fraction by $\frac{\sqrt{3}}{\sqrt{3}}$, and watch how the radical expression disappears from the denominator:$\displaystyle \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}$