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.

Key Insights

• 😑 Synthetic division and algebraic long division achieve the same result, simplifying expressions with divisors of the form "x + c" or "x - c".
• 🍉 Algebraic long division involves determining how many times the largest term in the divisor can be divided into the largest term in the dividend.
• ☺️ Synthetic division assumes a simplified divisor of just "x", making the process more efficient.
• 🍉 Coefficients in the dividend and divisor represent the same terms and can be manipulated accordingly.
• ➗ The process of multiplication and subtraction in algebraic long division and synthetic division is similar.
• ➗ Synthetic division uses less space and is faster compared to algebraic long division.
• ➗ Both synthetic division and algebraic long division allow for the simplification of polynomial expressions.

Transcript

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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.