Why synthetic division works  Polynomial and rational functions  Algebra II  Khan Academy  Summary and Q&A
TL;DR
Algebraic long division and synthetic division yield the same result with different approaches.
Questions & Answers
Q: How does algebraic long division differ from synthetic division?
Algebraic long division requires assessing how many times the highest degree term in the divisor goes into the highest degree term in the dividend. Synthetic division, on the other hand, assumes a simple divisor, like "x", to simplify the process.
Q: Why is it possible to drop coefficients straight down in synthetic division?
In synthetic division, dropping coefficients straight down is possible because the assumption is made that the divisor is "x" without any coefficients. Therefore, the coefficients in the dividend and the divisor in synthetic division remain the same.
Q: How do you multiply the divisor and quotient in algebraic long division?
To multiply the divisor and quotient in algebraic long division, distribute the terms of the divisor to each term in the quotient. Then, simplify the resulting expression.
Q: Why is the remainder the same in synthetic division and algebraic long division?
The remainder is the same in both synthetic division and algebraic long division because they both yield the same simplified expression. The remainder represents the unmatched terms that cannot be divided further.
Summary & Key Takeaways

Algebraic long division and synthetic division both aim to simplify the same expression.

To perform algebraic long division, determine how many times the highest degree term in the divisor goes into the highest degree term in the dividend.

The coefficients in the dividend and divisor represent the same terms and can be manipulated accordingly.