Can you get a remainder using synthetic division?

Yes, synthetic division is a method specifically designed to find both the quotient and the remainder when dividing a polynomial by a linear factor of the form (x - a).

Understanding Synthetic Division and Remainders

Synthetic division offers a streamlined way to perform polynomial division, especially when the divisor is linear. The process not only simplifies the division but also directly reveals the remainder.

• The Process: Synthetic division involves using the root of the divisor ( 'a' in 'x-a') and the coefficients of the dividend polynomial.
• The Result: After performing the synthetic division steps, the last number in the bottom row represents the remainder. The other numbers in the bottom row represent the coefficients of the quotient polynomial.

Representing the Remainder

The remainder obtained from synthetic division can be expressed in a couple of ways:

1. As a constant: The remainder is simply the numerical value resulting from the calculation. For example, if the result of synthetic division ends with the number 5, then 5 is the remainder.
2. With "R": You can explicitly indicate that the value is the remainder by writing "R" before it. For example, R = 5.
3. As a fraction: If further calculations are required, the remainder can be written as a fraction where the remainder is the numerator and the divisor (x-a) is the denominator. For instance, if the remainder is 5 and the divisor is (x - 2), the remainder term can be expressed as 5/(x - 2).

Example

Let's say we divide the polynomial x² + 3x + 2 by (x - 1) using synthetic division.

1. Set up the synthetic division with 1 (the root of x-1) and the coefficients 1, 3, and 2.

   1 | 1  3  2
     |    1  4
     |---------
       1  4  6

2. The last number, 6, is the remainder. The quotient is x + 4.

3. We can write the result as: x + 4 + 6/(x-1), where 6 is the remainder.

Conclusion

Synthetic division is a valuable tool for quickly determining both the quotient and the remainder of a polynomial division by a linear factor. The remainder is a direct result of the synthetic division process.