Division of polynomials

Definition

The division of polynomials is a method used to divide one polynomial by another, producing a quotient and possibly a remainder. This process can be performed similarly to numerical long division, where the goal is to simplify the polynomial fraction and find how many times the divisor fits into the dividend. Understanding how to divide polynomials is essential for working with rational functions and performing polynomial factorization.

5 Must Know Facts For Your Next Test

1. The process of dividing polynomials can be executed using either synthetic division or long division, depending on the context and the degree of the polynomials involved.
2. When dividing polynomials, if the degree of the divisor is greater than the degree of the dividend, the quotient is zero and the remainder is the dividend itself.
3. Polynomial division is a key step in finding roots of polynomials and simplifying rational expressions in algebra.
4. The remainder theorem states that if a polynomial $P(x)$ is divided by $x - c$, the remainder is $P(c)$, which can help in evaluating polynomials at specific points.
5. The division algorithm for polynomials guarantees that for any two polynomials $P(x)$ and $D(x)$ (with $D(x)$ not equal to zero), there exist unique polynomials $Q(x)$ (the quotient) and $R(x)$ (the remainder) such that $P(x) = D(x) imes Q(x) + R(x)$.

Review Questions

• How does the process of polynomial long division resemble numerical long division?
  • Polynomial long division follows a similar structure to numerical long division where you divide, multiply, and subtract step by step. You start with the leading term of the dividend and divide it by the leading term of the divisor to find the first term of the quotient. Then, you multiply this term by the entire divisor and subtract this product from the original polynomial. The steps continue until you either reach a polynomial of lower degree than your divisor or have no remainder.
• In what ways does understanding polynomial division assist in finding zeros or factors of polynomials?
  • Understanding polynomial division helps identify zeros or factors because when you divide a polynomial by a linear factor like $(x - c)$, if there's no remainder, then $(x - c)$ is a factor. Additionally, using synthetic division can simplify this process significantly when testing potential roots. If a root results in zero remainder, it confirms that value as a root and reveals other potential factors through further division.
• Evaluate how the remainder theorem relates to polynomial division and its applications in algebra.
  • The remainder theorem states that when dividing a polynomial $P(x)$ by $(x - c)$, the remainder equals $P(c)$. This relationship highlights an efficient way to evaluate polynomials at specific values without fully performing polynomial division. It provides a practical application in root finding, allowing us to quickly verify if $c$ is a root of $P(x)$ by simply computing $P(c)$. This theorem streamlines many algebraic processes including graph analysis and factoring.

Related terms

Polynomial: A mathematical expression consisting of variables, coefficients, and non-negative integer exponents combined using addition, subtraction, and multiplication.

Quotient: The result obtained from dividing one quantity by another, specifically in the context of polynomial division, where it represents the main outcome of the division process.

Remainder: The amount left over after dividing a polynomial by another polynomial that cannot be divided evenly; it provides insights into the division process.