Synthetic Division

Definition

Synthetic division is a shorthand method for dividing a polynomial by a linear expression, typically in the form of (x - a). This technique allows for efficient division of polynomials without the need for the full long division process, making it a valuable tool in the study of polynomial functions and their properties.

5 Must Know Facts For Your Next Test

1. Synthetic division is particularly useful when dividing a polynomial by a linear expression of the form $(x - a)$, where $a$ is a constant.
2. The process of synthetic division involves arranging the coefficients of the polynomial in a tabular format and performing a series of simple arithmetic operations to obtain the quotient and remainder.
3. The quotient obtained through synthetic division is a polynomial of degree one less than the original polynomial, and the remainder is a constant.
4. Synthetic division can be used to find the zeros (roots) of a polynomial function by determining the values of $x$ for which the remainder is zero.
5. The Remainder Theorem states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is equal to $P(a)$, which can be used to simplify the division process.

Review Questions

• Explain how synthetic division is used to divide a polynomial by a linear expression of the form $(x - a)$.
  • Synthetic division is a shorthand method for dividing a polynomial by a linear expression of the form $(x - a)$. The process involves arranging the coefficients of the polynomial in a tabular format and performing a series of simple arithmetic operations to obtain the quotient and remainder. The quotient is a polynomial of degree one less than the original polynomial, and the remainder is a constant. This technique is particularly useful when finding the zeros (roots) of a polynomial function, as the Remainder Theorem states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is equal to $P(a)$.
• Describe how synthetic division can be used to find the zeros (roots) of a polynomial function.
  • Synthetic division can be used to find the zeros (roots) of a polynomial function by determining the values of $x$ for which the remainder is zero. This is based on the Remainder Theorem, which states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is equal to $P(a)$. By repeatedly dividing the polynomial by linear expressions of the form $(x - a)$ and checking for a remainder of zero, you can identify the values of $x$ that make the polynomial equal to zero, which are the zeros (roots) of the function.
• Analyze the relationship between synthetic division and the Remainder Theorem, and explain how this connection can be leveraged to simplify the division process.
  • The Remainder Theorem is closely linked to the process of synthetic division. The theorem states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is equal to $P(a)$. This relationship can be leveraged to simplify the division process using synthetic division. By arranging the coefficients of the polynomial in a tabular format and performing a series of arithmetic operations, the quotient and remainder can be obtained efficiently, without the need for the full long division process. The Remainder Theorem ensures that the remainder obtained through synthetic division is equal to the value of the polynomial at the point $x = a$, which can be used to determine the zeros (roots) of the polynomial function.

Related terms

Polynomial Division:

The process of dividing one polynomial by another to obtain a quotient and a remainder.

Linear Expression: An expression in one variable that can be written in the form $ax + b$, where $a$ and $b$ are constants.

Remainder Theorem:

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