Long Division Example with Monic Linear Divisor

This page demonstrates the process of long division for polynomials using a monic linear divisor, highlighting the steps involved and introducing the concept of synthetic division.

D. Long Division Example

The page shows a detailed example of dividing 2x³ - 4x² + 7x - 11 by x + 2 using traditional long division.

Example: 2x³ - 4x² + 7x - 11 ÷ $x + 2$ = 2x² - 8x + 23 + 26/$x + 2$

The process is then simplified by focusing only on the coefficients and changing the divisor to "-2" to facilitate addition instead of subtraction.

Highlight: This simplification process is a step towards synthetic division, which further streamlines the calculation.

Synthetic Division

This page introduces the concept of synthetic division as a shortcut method for dividing polynomials by monic linear divisors.

E. Synthetic Division Process

The page demonstrates how to perform synthetic division using the same example from the previous page: 2x³ - 4x² + 7x - 11 ÷ $x + 2$.

Definition: Synthetic division is a simplified method for dividing polynomials by monic linear divisors of the form x - c.

Example: Dividing 2x³ - 4x² + 7x - 11 by x + 2 using synthetic division: -2) 2 -4 7 -11 -4 16 -46 2 -8 23 -57 Answer: 2x² - 8x + 23 + 26/$x + 2$

Highlight: Synthetic division only works for monic linear divisors. For all other cases, algebraic long division must be used.

Examples of Synthetic Division

This page provides two examples of synthetic division to reinforce understanding of the technique.

F. Examples

Example 1: Divide 3x³ - x + 5 by x - 1 using synthetic division

Example:

1. 3 0 -1 5 3 3 2 3 3 2 7 Answer: 3x² + 3x + 2 + 7/$x - 1$

Example 2: Divide 2x³ + 6x² + 5x - 3 by x + 3 using synthetic division

Example: -3) 2 6 5 -3 -6 0 -15 2 0 5 -18 Answer: 2x² + 5 - 18/$x + 3$

These examples demonstrate the efficiency of synthetic division compared to traditional long division methods for polynomials.

Highlight: Synthetic division is a powerful tool for quickly dividing polynomials by linear factors, making it invaluable for solving polynomial equations and finding roots.

3.3B Synthetic Division

This section introduces the concept of synthetic division, a simplified method for dividing polynomials by linear factors. It begins by defining key terms and progresses to demonstrate the technique through examples.

A. Linear Polynomials

Linear polynomials are defined as polynomials of the form ax + b, where a ≠ 0.

Example: 3x - 1, 2x - 5, and 3x - 1 are linear polynomials.

B. Monic Polynomials

Monic polynomials are polynomials with a leading coefficient of 1.

Example: x³ + 5x² - 2x - 1, x² - 3x² - 2, and x + b are monic polynomials.

C. Monic Linear Polynomials

Monic linear polynomials are both linear and monic, taking the form x - c.

Example: x + 5, x - 2, and x + 3 are monic linear polynomials.

Highlight: All monic linear polynomials can be written as x - c, where c can be positive, negative, or zero.