Divide Using Synthetic Division Calculator

Synthetic Division Method:

\[ \frac{P(x)}{(x - c)} = Q(x) + \frac{R}{(x - c)} \]

Quotient Polynomial:

Remainder:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Synthetic Division?

Synthetic division is a shorthand method of dividing a polynomial by a linear factor of the form (x - c). It's simpler and requires less writing than traditional polynomial long division.

2. How Does Synthetic Division Work?

The synthetic division algorithm:

Write the coefficients of the polynomial (highest degree first)
Write the constant c from (x - c) to the left
Bring down the first coefficient
Multiply by c and add to the next coefficient
Repeat until all coefficients are processed
The last number is the remainder
The other numbers are coefficients of the quotient polynomial

3. When to Use Synthetic Division

Details: Synthetic division can only be used when dividing by a linear factor (x - c). It's particularly useful for:

Evaluating polynomials (remainder theorem)
Factoring polynomials
Finding roots of polynomials

4. Using the Calculator

Tips:

Enter coefficients separated by commas (e.g., "2,-3,0,5" for 2x³ - 3x² + 5)
Include 0 for missing terms
Enter the constant c from (x - c)

5. Frequently Asked Questions (FAQ)

Q1: Can I use synthetic division for divisors other than (x - c)?
A: No, synthetic division only works for linear divisors of the form (x - c).

Q2: What does the remainder represent?
A: According to the Remainder Theorem, the remainder equals P(c), the polynomial evaluated at x = c.

Q3: How do I know if (x - c) is a factor?
A: If the remainder is 0, then (x - c) is a factor of the polynomial.

Q4: Can I use synthetic division for polynomials with missing terms?
A: Yes, but you must include 0 coefficients for the missing terms.

Q5: What's the advantage over long division?
A: Synthetic division is faster, requires less writing, and is less prone to errors for linear divisors.