key term - Long Division

Definition

Long division is a method for dividing a large number by another number, typically by breaking down the division process into a series of smaller steps. It is a systematic approach to division that allows for the calculation of quotients and remainders, even for complex numerical problems.

5 Must Know Facts For Your Next Test

1. Long division is a fundamental algorithm used to divide a large number by a smaller number, producing a quotient and a remainder.
2. The process of long division involves repeatedly subtracting the divisor from the dividend, keeping track of the number of times the divisor goes into the current portion of the dividend.
3. Partial fraction decomposition often involves the use of long division to break down a rational function into a sum of simpler rational functions.
4. Long division is a useful technique for solving problems involving polynomial division, which is an important concept in the study of rational functions.
5. The ability to perform long division fluently is a valuable skill for students studying calculus, as it underpins many of the techniques used in the course.

Review Questions

• Explain how long division is used in the context of partial fractions.
  • In the context of partial fractions, long division is often employed to break down a rational function into a sum of simpler rational functions. This is done by dividing the numerator polynomial by the denominator polynomial, with the resulting quotient and remainder being used to construct the partial fraction decomposition. The long division process allows for the identification of any linear or quadratic factors in the denominator, which are then used to determine the appropriate form of the partial fraction expansion.
• Describe the relationship between long division and polynomial division.
  • The process of long division for integers is closely related to the division of polynomials. Just as long division is used to divide a large integer by a smaller integer, polynomial division can be used to divide one polynomial by another. The steps involved in polynomial division mirror the steps of long division, with the divisor and dividend being polynomials rather than integers. This connection between long division and polynomial division is an important concept in the study of rational functions and partial fractions.
• Analyze the role of long division in the broader context of rational functions and their properties.
  • Long division is a fundamental tool in the study of rational functions, as it allows for the decomposition of complex rational expressions into simpler forms. This decomposition, known as partial fraction decomposition, is often a necessary step in evaluating integrals involving rational functions or solving differential equations. By using long division to break down a rational function, students can gain a deeper understanding of the structure and properties of rational functions, such as the behavior of the numerator and denominator, the presence of linear or quadratic factors, and the implications for the function's behavior and applications.

Related terms

Partial Fractions:

A technique used to express a rational function as a sum of simpler rational functions, which can be helpful in evaluating integrals and solving differential equations.

Polynomial Division:

The division of one polynomial by another, which can be performed using a process similar to long division for integers.

Rational Functions: A function that can be expressed as the ratio of two polynomials, which are often the focus of partial fraction decomposition.