Synthetic Division

Definition

Synthetic division is a shortcut method for dividing a polynomial by a linear expression of the form $(x - a)$. It allows for the efficient computation of the quotient and remainder without having to perform long division.

5 Must Know Facts For Your Next Test

1. Synthetic division is particularly useful when the divisor is a linear expression of the form $(x - a)$, where $a$ is a constant.
2. The process of synthetic division involves writing the coefficients of the dividend polynomial in a tabular form and performing a series of calculations to obtain the quotient and remainder.
3. The remainder obtained from synthetic division is equal to the value of the dividend polynomial when $x = a$, as stated by the Remainder Theorem.
4. If the remainder obtained from synthetic division is zero, then the linear expression $(x - a)$ is a factor of the dividend polynomial, as stated by the Factor Theorem.
5. Synthetic division can be used to find the roots of a polynomial equation by identifying the values of $a$ for which the remainder is zero.

Review Questions

• Explain how synthetic division is related to the Remainder Theorem.
  • Synthetic division is closely related to the Remainder Theorem, which states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is equal to $P(a)$. The process of synthetic division allows for the efficient computation of this remainder without having to perform long division. By writing the coefficients of the dividend polynomial in a tabular form and performing a series of calculations, the final value in the table represents the remainder, which is equal to the value of the polynomial when $x = a$.
• Describe how synthetic division can be used to determine the factors of a polynomial.
  • Synthetic division can be used to determine the factors of a polynomial by identifying the values of $a$ for which the remainder is zero. According to the Factor Theorem, if the remainder obtained from synthetic division is zero, then the linear expression $(x - a)$ is a factor of the dividend polynomial. By repeatedly applying synthetic division with different values of $a$, one can identify all the linear factors of the polynomial, which can then be used to factorize the polynomial completely.
• Analyze the relationship between synthetic division and the roots of a polynomial equation.
  • Synthetic division can be used to find the roots of a polynomial equation by identifying the values of $a$ for which the remainder is zero. Since the remainder obtained from synthetic division is equal to the value of the polynomial when $x = a$, if the remainder is zero, then $a$ is a root of the polynomial equation. By repeatedly applying synthetic division with different values of $a$, one can determine all the real roots of the polynomial equation. This connection between synthetic division and the roots of a polynomial equation is a powerful tool for solving polynomial equations.

Related terms

Polynomial Division:

The process of dividing one polynomial by another to obtain a quotient and a 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 $P(a) = 0$.