Algebra & Precalculus Tool
Synthetic Division Calculator
Divide polynomials of any degree by linear factors of the form (x - c) quickly and accurately. This tool generates the full synthetic division grid table, computes the final quotient polynomial and constant remainder, checks if the divisor represents a root of the equation, and displays factored forms step-by-step.
Synthetic Division Calculator
Divide a polynomial by a linear factor (x - c) step-by-step
* Comma or space separated. Enter 0 for missing exponents (e.g. x³ - 4x + 5 is 1, 0, -4, 5).
* Represents the c in (x - c). If dividing by (x - 3), enter 3. If dividing by (x + 2), enter -2.
Outputs
Enter polynomial coefficients and click Divide Polynomial.
What is Synthetic Division?
In algebra, Synthetic Division is a mathematical technique used to divide polynomials by linear divisors (binomials with degree 1). Unlike long division, which involves managing numerous variable terms and exponents, synthetic division streamlines the process by isolating the coefficients and performing basic addition and multiplication.
This method is highly prized in high school algebra and college precalculus courses because it allows students and mathematicians to quickly evaluate polynomials at specific values (via the Remainder Theorem), find factors and roots, and solve higher-degree polynomial equations.
Polynomial Long Division vs. Synthetic Division
While both methods achieve the same mathematical division result, they feature different structures, constraints, and speeds:
| Criteria | Polynomial Long Division | Synthetic Division |
|---|---|---|
| Divisor Limit | None (can divide by quadratic, cubic, etc.) | Linear divisors only (x - c) | Variables Handled | Explicitly writes variables and exponents | Only manipulates number coefficients |
| Speed & Efficiency | Slow, multiple writing steps | Extremely fast, minimal writing |
| Complexity/Error Risk | High risk of sign errors during subtraction | Low risk (relies on addition after sign flip) |
How to Perform Synthetic Division Manually
Follow this clear algorithm to divide a polynomial by a linear binomial $(x - c)$:
- Arrange terms: Write the dividend polynomial in standard form (descending order of exponents).
- Insert 0 placeholders: Add a zero coefficient for any skipped power of $x$ (e.g., $x^3 - x + 2$ has coefficients $1, 0, -1, 2$).
- Set up the grid: Write the root $c$ in a small box on the left, and list the dividend coefficients in a horizontal row on the right.
- Bring down: Copy the first coefficient straight down below the horizontal line.
- Multiply and add: Multiply the number below the line by the root $c$, write the result in the next column, and add the column numbers to find the next value below the line.
- Conclude: The final sum on the bottom right is the remainder. The remaining numbers represent the coefficients of the quotient polynomial (which is of degree $n-1$).
Benefits of Using the Synthetic Division Calculator
Worked Synthetic Division Examples
Example 1: Basic Division
Coefficients: 2, 5, -3, 7 | Divisor: (x - 3) | Root (c): 3
Original Coefficients: 2, 5, -3, 7
Root (c): 3
Step 1: Bring down the 2. (2)
Step 2: Multiply 2 by 3 = 6. Add to 5: 5 + 6 = 11. (11)
Step 3: Multiply 11 by 3 = 33. Add to -3: -3 + 33 = 30. (30)
Step 4: Multiply 30 by 3 = 90. Add to 7: 7 + 90 = 97. (97)
Quotient: 2x² + 11x + 30
Remainder: 97
Example 2: With Zero Coefficient (x⁴ - 4x² + 5)
Coefficients: 1, 0, -4, 0, 5 | Divisor: (x + 2) | Root (c): -2
Original Coefficients: 1, 0, -4, 0, 5 (places 0 for x³ and x)
Root (c): -2
Step 1: Bring down 1. (1)
Step 2: 1 × (-2) = -2. Add 0: 0 + (-2) = -2. (-2)
Step 3: -2 × (-2) = 4. Add -4: -4 + 4 = 0. (0)
Step 4: 0 × (-2) = 0. Add 0: 0 + 0 = 0. (0)
Step 5: 0 × (-2) = 0. Add 5: 5 + 0 = 5. (5)
Quotient: x³ - 2x²
Remainder: 5
Example 3: Finding Roots (x³ - 6x² + 11x - 6)
Coefficients: 1, -6, 11, -6 | Divisor: (x - 1) | Root (c): 1
Original Coefficients: 1, -6, 11, -6
Root (c): 1
Step 1: Bring down 1. (1)
Step 2: 1 × 1 = 1. Add -6: -6 + 1 = -5. (-5)
Step 3: -5 × 1 = -5. Add 11: 11 + (-5) = 6. (6)
Step 4: 6 × 1 = 6. Add -6: -6 + 6 = 0. (0)
Quotient: x² - 5x + 6
Remainder: 0 (Since R=0, x=1 is a root and (x-1) is a factor!)
Pro Tip: Solving Divisors with $ax - b$
If your divisor is in the form (ax - b) where a ≠ 1 (e.g., 2x - 3), you can still use synthetic division! First, factor out a to get a(x - b/a). Perform synthetic division using the root c = b/a. Remember that after dividing, you must divide the final quotient coefficients (but NOT the remainder) by a to get the correct quotient polynomial.
Frequently Asked Questions
- What is synthetic division?
- Synthetic division is a shorthand method of dividing a polynomial by a linear binomial of the form (x - c). It is much faster and requires less writing than traditional polynomial long division because it only uses the coefficients of the polynomial terms.
- When can you use synthetic division?
- You can use synthetic division when the divisor is a linear binomial of the form (x - c), where the coefficient of the x term is 1. If the divisor is quadratic (e.g., x² + 1) or has a leading coefficient other than 1 (e.g., 2x - 3), you must use polynomial long division instead, or modify the divisor first.
- How do you handle missing terms or powers in synthetic division?
- You must insert a coefficient of 0 for any missing power of x in the dividend. For example, if the dividend is x³ - 4x + 5, the coefficients are written as 1, 0, -4, 5. Forgetting to insert the 0 placeholder is the most common error in synthetic division and will yield incorrect results.
- What is the Remainder Theorem?
- The Remainder Theorem states that if a polynomial P(x) is divided by (x - c), the remainder obtained is equal to P(c). This makes synthetic division a very efficient way to evaluate a polynomial at a specific number.
- What is the Factor Theorem?
- The Factor Theorem states that a linear binomial (x - c) is a factor of a polynomial P(x) if and only if the remainder of dividing P(x) by (x - c) is zero. In other words, P(c) = 0.
- What happens if the divisor is (x + c) instead of (x - c)?
- If the divisor is (x + c), write it in the form (x - (-c)), which means the root/multiplier value you place in the synthetic division box is negative c (-c). For example, for a divisor of (x + 2), you divide using the root -2.
- How do you determine the degree of the quotient polynomial?
- The degree of the quotient polynomial is always exactly one less than the degree of the dividend polynomial. For example, if you divide a cubic polynomial (degree 3) by a linear divisor (degree 1), the quotient will be a quadratic polynomial (degree 2).
- Can synthetic division be used with fractional or decimal roots?
- Yes. The algorithm works exactly the same way with fractions, decimals, or negative numbers. However, performing calculations with fractions manually can be tedious, which is why using an online calculator is highly beneficial.