denominator

Examples of denominator in the following topics:

• Fractions Involving Radicals

  • Root rationalization is a process by which any roots in the denominator of an irrational fraction are eliminated.
  • In mathematics, we are often given terms in the form of fractions with radicals in the numerator and/or denominator.
  • 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.

• Rational Algebraic Expressions

  • The key is finding the least common denominator of the two rational expressions: the smallest multiple of both denominators.
  • Then, you rewrite the two fractions using this denominator.
  • If the two denominators are different, however, then you will need to use the above strategy of finding the least common denominator.
  • Notice the factors in the denominators.
  • Finding the prime factors of the denominators of two fractions enables us to find a common denominator.

• Fractions

  • To add fractions that contain unlike denominators (e.g. quarters and thirds), it is necessary to first convert all amounts to like quantities, which means all the fractions must have a common denominator.
  • One easy way to to find a denominator that will give you like quantities is simply to multiply together the two denominators of the fractions.
  • However, sometimes there is a faster way—a smaller denominator, or a least common denominator—that can be used.
  • For example, to add $\frac{3}{4}$ to $\frac{5}{12}$, the denominator 48 (the product of 4 and 12, the two denominators) can be used—but the smaller denominator 12 (the least common multiple of 4 and 12) may also be used.
  • Find a common denominator, and change each fraction to an equivalent fraction using that common denominator.

• Complex Fractions

  • A complex fraction is one in which the numerator, denominator, or both are fractions, which can contain variables, constants, or both.
  • A complex fraction, also called a complex rational expression, is one in which the numerator, denominator, or both are fractions.
  • Since there are no terms that can be combined or simplified in either the numerator or denominator, we'll skip to Step 3, dividing the numerator by the denominator:
  • Let's move on to Step 2: combine the terms in the denominator.
  • Let's turn to Step 3: divide the numerator by the denominator.

• Asymptotes

  • Vertical asymptotes occur only when the denominator is zero.
  • We can identify from the linear factors in the denominator that two singularities exist, at $x=1$ and $x = -1$.
  • Notice that, based on the linear factors in the denominator, singularities exists at $x=1$ and $x=-1$.
  • However, one linear factor $(x-1)$ remains in the denominator because it is squared.
  • The coefficient of the highest power term is $2$ in the numerator and $1$ in the denominator.

• Simplifying, Multiplying, and Dividing Rational Expressions

  • A rational expression is a fraction involving polynomials, where the polynomial in the denominator is not zero.
  • Recall that when two fractions are multiplied together, their numerators are multiplied to yield the numerator of their product, and their denominators are multiplied to yield the denominator of their product.
  • Following the rule for multiplying fractions, simply multiply their respective numerators and denominators:
  • Notice that we multiplied the numerators together and the denominators together, but we did not multiply the numerator by the denominator or vice-versa.
  • Multiplying out the numerator and denominator, this can be written as:

• Rational Equations

  • 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.
  • This suggests a strategy: Find a common denominator, set the numerators equal to each other, and solve for any unknowns.

• Introduction to Rational Functions

  • The domain of a rational function $f(x) = \frac{P(x)}{Q(x)}$ is the set of all values of $x$ for which the denominator $Q(x)$ is not zero.
  • Domain restrictions can be calculated by finding singularities, which are the $x$-values for which the denominator $Q(x)$ is zero.
  • Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational functions.
  • Singularity occurs when the denominator of a rational function equals $0$, whether or not the linear factor in the denominator cancels out with a linear factor in the numerator.
  • We can factor the denominator to find the singularities of the function:

• Complex Conjugates and Division

  • As shown earlier, c - di is the complex conjugate of the denominator c + di.
  • Neither the real part c nor the imaginary part d of the denominator can be equal to zero for division to be defined.
  • Practice dividing complex numbers by multiplying both the numerator and denominator by the complex conjugate of the denominator

• Partial Fractions

  • Partial fraction decomposition is a procedure used to reduce the degree of either the numerator or the denominator of a rational function.
  • Say we have a rational function $R(x) = \frac{f(x)}{g(x)}$, where the degree of the numerator is less than the degree of the denominator.
  • Assume $R(x)$ has a denominator that factors into other expressions, as $g(x)=P(x)\cdot Q(x)$, and that there are no repeated roots.
  • The first step to decomposing the function $R(x)$ is to factor its denominator:
  • To find a coefficient, multiply the denominator associated with it by the rational function $R(x)$: