Polynomial Division

Definition

Polynomial division is the process of dividing one polynomial by another to obtain a quotient and a remainder. It is a fundamental operation in algebra that allows for the factorization and simplification of polynomial expressions, which is crucial in understanding topics such as dividing polynomials and using partial fractions.

5 Must Know Facts For Your Next Test

1. Polynomial division follows a similar process to long division of integers, where the divisor is used to repeatedly subtract from the dividend until the remainder is smaller than the divisor.
2. The degree of the divisor must be less than or equal to the degree of the dividend for polynomial division to be possible.
3. Polynomial division is used to factor polynomials by finding common factors and to simplify rational expressions by reducing them to lowest terms.
4. The remainder theorem states that the remainder of dividing a polynomial $P(x)$ by $(x - a)$ is $P(a)$, which is useful in finding roots of polynomials.
5. Partial fraction decomposition, a technique used in calculus to integrate rational functions, relies on the principles of polynomial division.

Review Questions

• Explain the process of polynomial division and how it is used to factor polynomials.
  • Polynomial division involves dividing one polynomial by another to obtain a quotient and a remainder. The process is similar to long division of integers, where the divisor is used to repeatedly subtract from the dividend until the remainder is smaller than the divisor. Polynomial division is a crucial tool for factoring polynomials, as it allows you to find common factors and reduce the polynomial to a product of simpler factors. By dividing the polynomial and analyzing the quotient and remainder, you can identify the roots of the polynomial, which are the values of the variable that make the polynomial equal to zero.
• Describe the relationship between polynomial division and the remainder theorem, and how this can be used to find the roots of a polynomial.
  • The remainder theorem states that the remainder of dividing a polynomial $P(x)$ by $(x - a)$ is $P(a)$. This means that if the remainder of dividing a polynomial by $(x - a)$ is zero, then $a$ is a root of the polynomial. Conversely, if $a$ is a root of the polynomial, then the remainder of dividing the polynomial by $(x - a)$ will be zero. By using polynomial division and the remainder theorem, you can systematically find the roots of a polynomial, which is an important step in factoring and simplifying polynomial expressions.
• Explain how the concept of polynomial division is applied in the context of partial fraction decomposition, and discuss the significance of this technique in calculus.
  • Partial fraction decomposition is a technique used in calculus to integrate rational functions, which are fractions where both the numerator and denominator are polynomials. This process involves dividing the numerator polynomial by the denominator polynomial to obtain a quotient and a remainder, and then expressing the original rational function as a sum of simpler fractions. Polynomial division is a crucial step in this process, as it allows you to factor the denominator polynomial and identify the linear and quadratic factors that will be used to construct the partial fraction decomposition. The ability to integrate rational functions through partial fraction decomposition is essential in many areas of calculus, such as evaluating definite integrals and solving differential equations.

Related terms

Polynomial:

A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers.

Quotient:

The result obtained when one polynomial is divided by another, representing the part of the dividend that can be divided by the divisor.

Remainder:

The part of the dividend that cannot be divided by the divisor after performing polynomial division, which is left over as a result.