(x^2+1)/(x+1/4) Synthetic Division

3 min read Jun 17, 2024

Synthetic Division: A Step-by-Step Guide for (x^2 + 1) / (x + 1/4)

Synthetic division is a handy shortcut for dividing polynomials, especially when the divisor is a linear expression in the form (x - a). Let's break down how to perform synthetic division with the example (x² + 1) / (x + 1/4).

1. Set Up the Problem

Identify the coefficients: Write down the coefficients of the dividend (x² + 1), remembering to include any missing terms (in this case, the x term is missing). This gives us: 1, 0, 1.
Identify the divisor: We'll use the opposite of the constant term in the divisor (x + 1/4). So, our "a" is -1/4.

Let's arrange the setup:

-1/4 | 1   0   1 
      ----------------

2. Perform the Division

Bring down the first coefficient: Bring down the first coefficient (1) below the line.

-1/4 | 1   0   1 
      ----------------
          1

Multiply and add: Multiply the number you just brought down (1) by the divisor (-1/4), and write the result (-1/4) under the next coefficient (0). Add the two numbers.

-1/4 | 1   0   1 
      ----------------
          1   -1/4

Repeat the process: Multiply the new result (-1/4) by the divisor (-1/4) and write the product (1/16) under the next coefficient (1). Add the two numbers.

-1/4 | 1   0   1 
      ----------------
          1   -1/4  17/16

3. Interpret the Results

The numbers below the line represent the coefficients of the quotient, starting with the term with one less power than the dividend. The final number is the remainder.

Therefore, the result of the division is:

(x² + 1) / (x + 1/4) = x - 1/4 + 17/16 / (x + 1/4)

Or, in more familiar form:

(x² + 1) / (x + 1/4) = x - 1/4 + (17/16) / (x + 1/4)

Conclusion

Synthetic division provides a concise and efficient method to divide polynomials, particularly when dealing with linear divisors. By understanding the steps involved, you can effectively solve such divisions and gain valuable insights into the relationships between polynomials.