How Does Synthetic Division Work?

Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form x - k. It's a shorthand approach to polynomial long division and works by focusing on the coefficients of the polynomial and the value k. Here's a detailed breakdown of how it works:

The Setup

Before you start, make sure the polynomial is written in descending order of exponents, including terms with zero coefficients as placeholders. For example, if you have x³ + 2x + 1, you would write it as x³ + 0x² + 2x + 1. Then, follow these steps:

1. Write Down the Coefficients: Identify the coefficients of the polynomial. Write these in a row.
2. Set up the divisor: Write the value k from the linear divisor x - k to the left.
3. Draw a line: Underneath the row of coefficients, draw a horizontal line.

Example: For (x³ + 2x² - 5x + 3) ÷ (x - 2), we set it up like this:

2	1	2	-5	3
---	---	---	---	---

The Process

The core process of synthetic division involves these iterative steps, as outlined in the reference:

1. Bring Down: Bring the leading coefficient down below the horizontal line.

   2	1	2	-5	3
   ---	---	---	---	---
   	1

2. Multiply and Add: Multiply the number you just brought down by the value to the left (k) and place the product in the next column below the coefficients of the polynomial. Add this product to the coefficient above it, placing the sum below the line.

   2	1	2	-5	3
   		2
   ---	---	---	---	---
   	1	4

3. Repeat: Continue the process of multiplying the last entry below the line by k (2 in this case) and add to the next term above until you reach the end of the polynomial coefficient row.

   2	1	2	-5	3
   		2	8
   ---	---	---	---	---
   	1	4	3

   2	1	2	-5	3
   		2	8	6
   ---	---	---	---	---
   	1	4	3	9

4. Interpret Results: The last number below the line is the remainder, and all the other numbers are the coefficients of the quotient.

Example: Interpreting the Result

Looking at our completed synthetic division:

2	1	2	-5	3
		2	8	6
---	---	---	---	---
	1	4	3	9

We see the numbers 1, 4, and 3. These are the coefficients of our quotient, and the last number, 9, is the remainder. Since the original polynomial was x³, the quotient will be one degree lower, x². Therefore, we have:

• Quotient: 1x² + 4x + 3
• Remainder: 9

So, (x³ + 2x² - 5x + 3) ÷ (x - 2) = x² + 4x + 3 + 9/(x-2)

Practical Insights

• Synthetic division offers a faster way to divide a polynomial by a linear binomial.
• It is critical to include place holders with a 0 coefficient if any power of x is missing
• The remainder theorem, which states that when a polynomial f(x) is divided by x - k, the remainder is f(k), can be easily verified using synthetic division.

In summary, synthetic division is a simple, efficient method for dividing polynomials by a linear factor. It is a core technique in algebra and is incredibly useful in further studies of calculus.