Polynomial Long Division

Definition

Polynomial long division is a method used to divide a polynomial by another polynomial, similar to the long division algorithm used for dividing integers. This technique allows for the division of polynomials and the determination of the quotient and remainder.

5 Must Know Facts For Your Next Test

1. Polynomial long division is a step-by-step process that involves dividing the leading term of the dividend by the leading term of the divisor, and then subtracting the resulting multiple of the divisor from the dividend.
2. The process continues until the degree of the remainder is less than the degree of the divisor, or until the remainder is zero, indicating that the divisor is a factor of the dividend.
3. The quotient obtained from the polynomial long division process is the result of the division, and the remainder (if any) is the value that is left over.
4. Polynomial long division is useful for solving polynomial equations, factoring polynomials, and finding the roots of polynomial functions.
5. The Remainder Theorem and the Factor Theorem are closely related to polynomial long division and can be used to simplify the division process.

Review Questions

• Explain the step-by-step process of performing polynomial long division.
  • To perform polynomial long division, the first step is to divide the leading term of the dividend by the leading term of the divisor. This gives the first term of the quotient. Next, multiply the divisor by the first term of the quotient and subtract the result from the dividend. This gives the first term of the remainder. The process is then repeated with the remainder as the new dividend, until the degree of the remainder is less than the degree of the divisor, or until the remainder is zero. The final quotient and remainder are the results of the polynomial long division.
• Describe the relationship between polynomial long division and the Remainder Theorem.
  • The Remainder Theorem states that when a polynomial $P(x)$ is divided by $(x - a)$, the remainder is equal to $P(a)$. This means that the remainder obtained from the polynomial long division process is the value of the polynomial when $x = a$. This relationship can be used to simplify the division process and to determine the roots of polynomial equations.
• Analyze how polynomial long division can be used to factor polynomials.
  • Polynomial long division can be used to factor polynomials by finding the factors of the divisor that make the remainder zero. If the remainder is zero, then the divisor is a factor of the dividend. This process can be repeated to find all the factors of the polynomial. Additionally, the quotient obtained from the polynomial long division process can be used to identify other factors of the polynomial, as the quotient and the divisor together form the factorization of the original polynomial.

Related terms

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

Remainder Theorem: The Remainder Theorem states that when a polynomial $P(x)$ is divided by $(x - a)$, the remainder is equal to $P(a)$.

Factor Theorem: The Factor Theorem states that a polynomial $P(x)$ is divisible by $(x - a)$ if and only if $P(a) = 0$.