Synthetic Division Calculator
Fast polynomial division by (x - c) with step-by-step synthetic division process
Enter polynomial in standard form
For (x - 2), enter 2. For (x + 3), enter -3
Understanding Synthetic Division
Synthetic division is a shorthand method for dividing a polynomial by a linear factor of the form (x - c). It's much faster than long division and requires less writing, making it ideal for quick calculations and finding polynomial roots.
When to Use Synthetic Division
Use synthetic division only when dividing by a linear factor (x - c). If your divisor is (x - 2), (x + 5), or any first-degree polynomial, synthetic division is perfect. For divisors of degree 2 or higher, you must use polynomial long division instead.
Steps for Synthetic Division
- Set Up: Write coefficients in order, including 0 for missing terms
- Identify c: If dividing by (x - c), use c. For (x + 3), use -3
- Bring Down: Copy the first coefficient to the bottom row
- Multiply and Add: Multiply bottom number by c, add to next coefficient
- Repeat: Continue multiplying and adding across all coefficients
- Read Result: Bottom row gives quotient coefficients and remainder (last number)
Example Walkthrough
Divide x³ - 6x² + 11x - 6 by (x - 2):
- Coefficients: [1, -6, 11, -6]
- Using c = 2
- Bring down 1
- 1 × 2 = 2, add to -6 = -4
- -4 × 2 = -8, add to 11 = 3
- 3 × 2 = 6, add to -6 = 0
- Result: x² - 4x + 3 with remainder 0
Advantages of Synthetic Division
- Much faster than long division
- Requires less space and writing
- Fewer opportunities for arithmetic errors
- Easy to use for testing potential roots
- Perfect for repeated divisions when factoring
Frequently Asked Questions
What's the difference between synthetic division and long division?
Synthetic division is a shortcut method that only works when dividing by (x - c). It's faster and requires less writing. Long division works for any divisor but takes more time and space. Use synthetic division whenever possible for speed.
How do I handle (x + 3) as a divisor?
Rewrite (x + 3) as (x - (-3)). So for synthetic division, use c = -3. Remember: the sign of c is opposite to the sign in the divisor. For (x - 5), use c = 5. For (x + 7), use c = -7.
What if my polynomial is missing terms?
Include 0 as the coefficient for any missing terms. For example, x³ + 5 should be written with coefficients [1, 0, 0, 5] to account for the missing x² and x terms. This keeps your work properly aligned.
What does a remainder of 0 mean?
If the remainder is 0, it means (x - c) is a factor of the polynomial, and c is a root (or zero) of the polynomial. This is the Factor Theorem and Remainder Theorem in action. You can use this to find all roots of a polynomial.
Can I use synthetic division to factor polynomials?
Yes! Test potential roots using synthetic division. If you get a remainder of 0, you've found a factor. You can then use synthetic division on the quotient to find more factors. Repeat until the polynomial is fully factored.
How do I verify my answer?
Multiply the quotient by (x - c) and add the remainder. You should get the original polynomial. In formula: (Quotient)(x - c) + Remainder = Original Polynomial. This verification works the same as for any division.
Can synthetic division work with fractions?
Yes! You can use synthetic division with any real number for c, including fractions and decimals. The process is exactly the same. For example, you can divide by (x - 1/2) using c = 0.5 or c = 1/2.
What degree will the quotient be?
The quotient will always have degree one less than the original polynomial. If you divide a cubic (degree 3) by a linear factor, you get a quadratic (degree 2). This is because you're dividing by a degree 1 polynomial.
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