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

4 min read Jun 17, 2024

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

Synthetic division is a handy shortcut for dividing polynomials by binomials of the form (x - a). It simplifies the long division process and makes it more efficient. Let's apply this method to divide (x^4 + 1) by (x + 1).

Setting Up the Problem

Identify the coefficients: In the polynomial (x^4 + 1), we have coefficients of 1, 0, 0, 0, and 1 for the terms x^4, x^3, x^2, x, and the constant term, respectively.
Find the divisor: We are dividing by (x + 1). Since the binomial is in the form (x - a), a = -1.

Set up the synthetic division table:

    -1 | 1  0  0  0  1 
         ----------------

Performing the Division

Bring down the leading coefficient: Bring down the first coefficient, 1, below the line.
```
    -1 | 1  0  0  0  1 
         ----------------
           1
```
Multiply and add: Multiply the number you just brought down (1) by the divisor (-1) and write the product (-1) under the next coefficient (0). Add the two numbers and write the sum (-1) below the line.
```
    -1 | 1  0  0  0  1 
         ----------------
           1 -1
```

Repeat the process: Repeat steps 2 and 3 for the remaining coefficients.

    -1 | 1  0  0  0  1 
         ----------------
           1 -1  1 -1  2

Interpreting the Results

The numbers below the line represent the coefficients of the quotient and the remainder. Therefore:

Quotient: x^3 - x^2 + x - 1
Remainder: 2

Final Result

We can express the result of the division as follows:

(x^4 + 1) / (x + 1) = x^3 - x^2 + x - 1 + 2/(x + 1)

This means that (x^4 + 1) divided by (x + 1) is equal to x^3 - x^2 + x - 1 with a remainder of 2.

Benefits of Synthetic Division

Efficiency: Synthetic division is quicker and less prone to errors than long division.
Clarity: The table format helps visualize the steps and organize the calculations.
Flexibility: Synthetic division can be used to find roots of polynomials and to evaluate polynomial functions at specific values.

By understanding the steps and the interpretation of the results, you can efficiently use synthetic division to divide polynomials and gain valuable insights into their properties.