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Synthetic division is a simplified method for dividing polynomials where the divisor is in the form of \(x - c\). It streamlines the traditional long division process. Here’s how it works step-by-step:
First, identify the root of the divisor. For example, if you have \(b + 1\), the root is \(-1\).
Next, write the coefficients of the polynomial you are dividing. In our example, these are 3 for \(3b^3\), 1 for \(b^2\), 0 for the missing \(b\) term, and 2 for the constant. All terms must be represented.
Perform synthetic division following these smaller steps which involve multiplying and adding, instead of traditional polynomial division.
This method significantly reduces the complexity and helps in quickly finding the quotient and remainder.

The quotient polynomial is the result of the division of two polynomials, excluding the remainder. After performing synthetic division, the sequence of numbers that you obtain below the line represents the coefficients of your quotient.
In the exercise given, the coefficients are found through the process:

• Drop down the first coefficient (3)
• Then, multiply each result by the root, and add to the next coefficient

This forms a new polynomial: \(3b^2 - 2b + 2\). Thus, the quotient polynomial is what's left after dividing the higher-degree polynomial by the divisor.

The remainder is what remains after the division process, and it’s the last value written below the line in synthetic division. This value indicates how much is left over after the polynomial has been divided as completely as possible by the divisor.
In our example, the last number we write down during the synthetic division process is 0. Hence, the division of \(3b^3 + b^2 + 2\) by \(b+1\) does not leave any remainder.
If a non-zero remainder is present, it would be written as a fraction alongside the quotient. For example, if the remainder was 3, it would be written as \( \frac{3}{b+1} \). Understanding and recognizing the remainder helps in fully comprehending the result of your polynomial division.