Synthetic Division

Definition

Synthetic division is a shortcut method for dividing a polynomial by a linear expression of the form (x - a), where 'a' is a constant. It allows for the efficient computation of polynomial division without the need for long division, making it a valuable tool in the context of dividing polynomials and solving polynomial equations.

5 Must Know Facts For Your Next Test

1. Synthetic division is a simplified version of long division, where the divisor is a linear expression of the form (x - a).
2. The process of synthetic division involves arranging the coefficients of the dividend polynomial in a specific pattern and performing a series of arithmetic operations to obtain the quotient and remainder.
3. Synthetic division is particularly useful when the divisor is a linear expression with a constant term, as it allows for the efficient computation of polynomial division.
4. The remainder obtained through synthetic division is equal to the value of the dividend polynomial when x is replaced by the constant term of the divisor, as per the Remainder Theorem.
5. The Factor Theorem states that a linear expression (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0, which can be easily determined using synthetic division.

Review Questions

• Explain how synthetic division simplifies the process of dividing polynomials.
  • Synthetic division is a shortcut method for dividing a polynomial by a linear expression of the form (x - a), where 'a' is a constant. It eliminates the need for the long division algorithm, which can be cumbersome, especially for higher-degree polynomials. By arranging the coefficients of the dividend in a specific pattern and performing a series of arithmetic operations, synthetic division allows for the efficient computation of the quotient and remainder, making it a valuable tool in the context of dividing polynomials.
• Describe the relationship between synthetic division and the Remainder Theorem.
  • The Remainder Theorem states that the remainder when a polynomial P(x) is divided by (x - a) is equal to P(a). This property is directly reflected in the process of synthetic division, where the final remainder obtained is the value of the dividend polynomial when x is replaced by the constant term of the divisor. This connection between synthetic division and the Remainder Theorem makes it a powerful technique for determining the value of a polynomial at a specific point, which is crucial in the context of solving polynomial equations.
• Explain how synthetic division can be used to determine the factors of a polynomial, as per the Factor Theorem.
  • The Factor Theorem states that a linear expression (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0. This relationship can be easily verified using synthetic division. By performing synthetic division with the divisor (x - a), if the final remainder is zero, then the linear expression (x - a) is a factor of the polynomial. Conversely, if the final remainder is non-zero, then (x - a) is not a factor of the polynomial. This application of synthetic division is particularly useful in the context of solving polynomial equations, as it allows for the identification of the roots of the polynomial, which correspond to the factors of the form (x - a).

Related terms

Polynomial Division: The process of dividing one polynomial by another to find the quotient and remainder.

Remainder Theorem:

A theorem that states that the remainder when a polynomial P(x) is divided by (x - a) is equal to P(a).

Factor Theorem:

A theorem that states that a linear expression (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0.