1. Long Division of Polynomials

Concept: Long division mirrors numeric long division. Divide the leading terms, multiply the divisor by that result, subtract, and repeat.

Steps

1. Arrange terms in descending powers of x. Fill missing terms with coefficient 0.
2. Divide the leading term of the dividend by the leading term of the divisor.
3. Multiply the divisor by this quotient term and subtract from the dividend.
4. Repeat until the remainder has lower degree than the divisor.

Example

Divide 2x³ + 3x² − 5x + 6 by x − 2.

2. Synthetic Division (Quick Method)

Concept: Synthetic division works directly with coefficients and is faster when dividing by x − a. It cannot be used for divisors with degree > 1 or leading coefficient ≠ 1 without modification.

Steps

1. Write the coefficients of the dividend.
2. Write the value a (the zero of the divisor x − a).
3. Bring down the first coefficient.
4. Multiply it by a, write under the next coefficient, add, and repeat.

Example (same as above)

Divide 2x³ + 3x² − 5x + 6 by x − 2 (so a = 2).

Tip: Always include zero coefficients for missing degrees. Synthetic division is great for quick root testing using the Remainder Theorem.

When to Use Which Method

Remainder & Factor Theorems

Remainder Theorem: When f(x) is divided by x − a, the remainder equals f(a).

Factor Theorem: If f(a) = 0, then x − a is a factor of f(x).

Practice Questions

1. Divide x³ − 4x² + 5x − 2 by x − 3 using synthetic division.
2. Divide 3x⁴ − 2x³ + 4x − 8 by x² − 2 using long division.
3. Use synthetic division to find the remainder when 2x³ − 3x² + 4x − 5 is divided by x + 2.
4. Verify answers using the Remainder Theorem.

Conclusion

Synthetic division is a powerful, time-saving shortcut for dividing by linear expressions of the form x − a. Long division remains necessary for higher-degree divisors or when the divisor's leading coefficient is not 1. Use the method that fits the divisor and the task.