Synthetic Division Calculator
Divide polynomials by linear factors (x - c) using synthetic division with step-by-step solutions
P(x) ÷ (x - c) = Q(x) + R/(x - c)
Include 0 for missing terms (e.g., 1, 0, -4 for x² - 4)
For (x + 3), enter -3
Please enter at least 2 coefficients for the polynomial.
÷
Quotient Q(x)
Remainder R
✓ is a factor!
Complete Division Result
P(x) = × () +
Synthetic Division Table
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Step-by-Step Solution
Step
Result: Quotient = , Remainder =
Verification
To verify: Multiply the quotient by the divisor and add the remainder. You should get the original polynomial.
× () + =
Factor Found!
Since the remainder is 0:
- • is a factor of the polynomial
- • x = is a root of the polynomial
- • The polynomial can be factored as: ()
The Synthetic Division Method
Algorithm Steps
- 1. Write the divisor value (c) and all coefficients
- 2. Bring down the first coefficient
- 3. Multiply by c, write under next coefficient
- 4. Add the column, repeat steps 3-4
- 5. Last number is remainder, others form quotient
Key Theorems
Remainder Theorem
P(c) equals the remainder when P(x) is divided by (x - c)
Factor Theorem
(x - c) is a factor of P(x) if and only if P(c) = 0
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About Synthetic Division Calculator
What is Synthetic Division?
Synthetic Division is a shorthand method for dividing a polynomial by a linear factor of the form (x - c). It's significantly faster than polynomial long division because it works only with the coefficients, avoiding the need to write out variable terms during calculations.
How to Use This Calculator
- Enter Polynomial Coefficients: Enter the coefficients of the dividend polynomial in descending order of powers, separated by commas (e.g., "1, -5, 6" for x² - 5x + 6)
- Enter Divisor Value (c): Enter the 'c' value from the divisor (x - c). For (x - 3), enter "3". For (x + 2), enter "-2"
- Calculate: View the complete step-by-step solution, quotient polynomial, and remainder
The Formula Behind the Calculation
Synthetic division performs the algebraic division represented by the formula:
P(x) / (x - c) = Q(x) + R / (x - c)
Where:
- P(x) is the dividend polynomial
- (x - c) is the linear divisor
- Q(x) is the quotient polynomial
- R is the remainder
Understanding Your Results
Quotient Polynomial
The resulting polynomial has a degree one less than the original dividend. The coefficients from synthetic division form this quotient.
Remainder
If the remainder is 0, it means (x - c) is a factor of the polynomial, and c is a root.
The Remainder Theorem
The remainder when dividing P(x) by (x - c) equals P(c). This connection makes synthetic division useful for evaluating polynomials.
FAQ (Frequently Asked Questions)
What if I have a missing term in my polynomial?
Use 0 as a placeholder. For x³ + 4x - 1 (missing x² term), enter: 1, 0, 4, -1
How do I handle (x + c) as a divisor?
For (x + c), use -c in synthetic division. For example, for (x + 3), use c = -3.
Can I verify my answer?
Yes! Multiply the quotient by (x - c) and add the remainder. You should get the original polynomial.
Important Limitations
- The divisor must be linear (degree 1)
- The leading coefficient of the divisor must be 1
- For divisors like (2x - 3), first convert to 2(x - 1.5)
Disclaimer: This calculator is for informational purposes only and should not replace professional educational guidance. Always verify complex calculations manually for critical academic work.