Explain the process of synthetic division in polynomials.

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

To perform synthetic division, the divisor (the linear factor) is written in the form (x-a), where a is the constant that makes the factor equal to zero. The coefficients of the polynomial are then written in a row, with any missing terms represented by a coefficient of zero. The first coefficient is brought down and multiplied by a, and the result is written below the second coefficient. The sum of the second coefficient and the result is then multiplied by a, and the result is written below the third coefficient. This process is repeated until all coefficients have been brought down and multiplied by a.

The final row of numbers represents the coefficients of the quotient polynomial, with the constant term being the last number in the row. If the remainder is zero, then the divisor is a factor of the polynomial. If there is a non-zero remainder, then the quotient polynomial and the remainder can be written as a fraction.

For example, to divide the polynomial 2x^3 + 5x^2 - 3x - 2 by the linear factor (x-1), we first write the divisor as (x-1) and a as 1. The coefficients of the polynomial are then written in a row:

1 | 2 5 -3 -2

The first coefficient is brought down and multiplied by 1:

1 | 2 5 -3 -2
2

The result is written below the second coefficient, and the sum is multiplied by 1:

1 | 2 5 -3 -2
2 7

The result is written below the third coefficient, and the sum is multiplied by 1:

1 | 2 5 -3 -2
2 7 4

The final row represents the coefficients of the quotient polynomial, which is 2x^2 + 7x + 4. The remainder is -6, so the division can be written as:

2x^3 + 5x^2 - 3x - 2 = (x-1)(2x^2 + 7x + 4) - 6