Use Synthetic Division to Solve x^4 + x + 1: What is the Quotient?

Synthetic division is a simplified method for dividing polynomials. In this case, we are tasked with dividing the polynomial x⁴ + x + 1 by a linear factor.

For synthetic division, we typically need to know the factor we are dividing by. However, since that information is not provided in your question, let’s assume we want to divide our polynomial by x – 1 for illustration purposes.

To perform synthetic division using x – 1, we use the root of the factor, which is x = 1. Here are the steps:

1. Write down the coefficients of the polynomial: 1, 0, 0, 1, 1 (these correspond to x⁴, x³, x², x¹, x⁰ respectively).
2. Bring down the leading coefficient (1).
3. Multiply this by the root (1) and write the result under the next coefficient (0).
4. Add down the column: 0 + 1 = 1. Repeat this step until you reach the last coefficient.

The process looks like this:

1 | 1  0  0  1  1
  |    1  1  1  2
-------------------
    1  1  1  2  3

The last value (3) is the remainder, while the values in the bottom row (1, 1, 1, 2) represent the coefficients of the quotient polynomial.

So, the quotient is:

x³ + x² + x + 2 with a remainder of 3.

Hence, when dividing x⁴ + x + 1 by x – 1, we get:

Quotient: x³ + x² + x + 2

Remainder: 3