Explain the process of synthetic division.

Synthetic division is a method used to divide a polynomial by a linear factor.

To perform synthetic division, we first write the polynomial in descending order of powers. Then, we write the divisor in the form (x-a), where a is the root of the divisor. We place the root a outside the division symbol and write the coefficients of the polynomial inside the division symbol. We then draw a line underneath the coefficients.

Next, we bring down the first coefficient and multiply it by the root a. We write the result underneath the second coefficient and add it to the second coefficient. We repeat this process until we reach the end of the polynomial. The final result is the quotient of the division, with the remainder written as a fraction over the divisor.

For example, let's divide the polynomial 2x^3 + 5x^2 - 3x - 2 by the linear factor (x-1). We first write the polynomial in descending order of powers:

2x^3 + 5x^2 - 3x - 2

We then write the divisor in the form (x-1) and place the root 1 outside the division symbol:

1 | 2 5 -3 -2

We bring down the first coefficient, which is 2, and multiply it by the root 1 to get 2. We write this underneath the second coefficient and add it to get 7. We repeat this process:

1 | 2 5 -3 -2
| 2 7 4
|__________
2 7 4 2

The final result is 2x^2 + 7x + 4, with a remainder of 2/(x-1).