Synthetic Division Calculator

Divide polynomials by a linear factor of the form (x - c)

Enter the coefficients of your polynomial and the constant 'c' from the divisor to find the quotient and remainder.

Examples

Click on an example to load it into the calculator.

Basic Division

example

Divide x² + 5x + 6 by x + 2 (c = -2)

P(x): 1, 5, 6

c: -2

Division with Remainder

example

Divide 2x³ - 3x² + 4x - 1 by x - 1 (c = 1)

P(x): 2, -3, 4, -1

c: 1

Missing Term

example

Divide x⁴ - 16 by x - 2 (c = 2). Note the zero coefficients for missing terms.

P(x): 1, 0, 0, 0, -16

c: 2

Division by (x + a)

example

Divide 3x³ + 2x² - x + 8 by x + 3 (c = -3)

P(x): 3, 2, -1, 8

c: -3

Other Titles
Understanding Synthetic Division: A Comprehensive Guide
Explore the principles, steps, and applications of synthetic division, a streamlined method for dividing polynomials.

What is Synthetic Division? Core Concepts

  • A shortcut for polynomial division by a linear binomial (x - c)
  • Based on the Remainder Theorem and Factor Theorem
  • Simplifies the long division process by using only coefficients
Synthetic division is an elegant, simplified method for dividing a polynomial by a linear binomial of the form (x - c). It streamlines the traditional long division process by eliminating the need to write out variables, focusing solely on the polynomial's coefficients. This makes it a faster and less error-prone technique, widely used in algebra to find roots, factor polynomials, and evaluate polynomial expressions.
The Logic Behind the Shortcut
The method is built upon the principles of the Remainder Theorem, which states that if a polynomial P(x) is divided by (x - c), the remainder is P(c). Synthetic division not only finds this remainder quickly but also determines the coefficients of the quotient polynomial. If the remainder is zero, it confirms that (x - c) is a factor of the polynomial, a concept known as the Factor Theorem.

Key Principles

  • Dividing P(x) by (x - c) gives a quotient Q(x) and a remainder R.
  • The relationship is P(x) = (x - c)Q(x) + R.
  • If R = 0, then 'c' is a root of the polynomial.

Step-by-Step Guide to Using the Synthetic Division Calculator

  • Correctly format polynomial coefficients
  • Determine the divisor constant 'c'
  • Interpret the quotient and remainder from the results
Our calculator simplifies the synthetic division process. Follow these steps for an accurate calculation.
1. Enter Polynomial Coefficients
In the 'Polynomial Coefficients' field, enter the coefficients of the polynomial you want to divide. Separate them with commas or spaces. Crucially, you must include a '0' for any missing terms in descending order of power. For example, for the polynomial P(x) = 2x⁴ - x² + 5, the correct input is '2, 0, -1, 0, 5'.
2. Enter the Divisor Constant (c)
In the 'Divisor Constant (c)' field, enter the value of 'c' from your divisor (x - c). Remember: if you are dividing by (x - 4), c is 4. If you are dividing by (x + 3), which is equivalent to (x - (-3)), then c is -3.
3. Interpret the Results
The calculator will provide two outputs: the 'Quotient (Q(x))' and the 'Remainder (R)'. The quotient field will show the coefficients of the resulting polynomial, which will be one degree lower than the original. The remainder will be a single constant value.

Practical Usage Examples

  • Input: P(x) = '1, -3, -10', c = '5' -> Output: Q(x) = [1, 2], R = 0. (x-5 is a factor)
  • Input: P(x) = '1, 0, -4, 1', c = '2' -> Output: Q(x) = [1, 2, 0], R = 1.

Real-World Applications of Synthetic Division

  • Finding roots of higher-degree polynomials in engineering
  • Factoring polynomials for mathematical modeling
  • Used in computer graphics for curve and surface calculations
Synthetic division is not just an academic exercise; it has practical applications in various scientific and technical fields.
Engineering and Physics
Engineers often encounter higher-degree polynomial equations when analyzing system stability, circuits, and mechanical vibrations. Synthetic division provides a quick method to test for rational roots, which is a crucial first step in solving these complex equations.
Computer Science and Graphics
In computer-aided design (CAD) and graphics, polynomials define curves and surfaces (like Bézier curves). Evaluating and manipulating these polynomial representations often involves techniques related to synthetic division for computational efficiency.
Economics and Finance
Financial models can sometimes involve polynomials to forecast trends or calculate complex interest scenarios. Synthetic division can be used to analyze the behavior of these models at specific points.

Industry Applications

  • Stability analysis in control systems theory.
  • Solving for eigenvalues in linear algebra.
  • Cryptographic algorithms that rely on polynomial factorization.

Common Misconceptions and Correct Methods

  • Forgetting to include zero coefficients for missing terms
  • Using the wrong sign for the constant 'c'
  • Misinterpreting the degree of the quotient polynomial
Mistake 1: Omitting Zeros for Missing Terms
A very common error is to forget to include placeholders for terms with a coefficient of zero. For a polynomial like x³ - 2x + 1, you must write its coefficients as '1, 0, -2, 1'. Leaving out the zero for the x² term will lead to an incorrect result.
Mistake 2: Incorrect Sign for 'c'
The divisor must be in the form (x - c). If you need to divide by (x + 5), you must rewrite it as (x - (-5)) and use c = -5 in your calculation. Using c = 5 would be incorrect and would be equivalent to dividing by (x - 5).
Mistake 3: The Degree of the Quotient
Remember that the degree of the quotient polynomial is always one less than the degree of the original dividend polynomial. If you divide a 4th-degree polynomial, your quotient will start with an x³ term.

Avoiding Common Pitfalls

  • Correct for x⁴ - 1: '1, 0, 0, 0, -1'
  • Incorrect for x⁴ - 1: '1, -1'
  • Correct for division by (x+7): c = -7
  • Incorrect for division by (x+7): c = 7

The Mathematical Steps of Synthetic Division

  • A breakdown of the algorithm's mechanics
  • How coefficients are multiplied and added in sequence
  • Connecting the final row of numbers to the quotient and remainder
Let's perform a manual synthetic division for P(x) = x³ - 7x - 6 divided by (x + 2). Here, c = -2.
Setup:
  1. Write the constant 'c' (-2) to the left.
  2. Write the coefficients of the polynomial (1, 0, -7, -6) to the right.
-2 | 1   0   -7   -6
   |____________
Execution:
  1. Bring Down: Drop the first coefficient (1) down. -2 | 1 0 -7 -6 |____________ 1
  2. Multiply and Add: Multiply the dropped value (1) by 'c' (-2), which is -2. Place it under the next coefficient (0) and add. 0 + (-2) = -2. -2 | 1 0 -7 -6 | -2 |____________ 1 -2
  3. Repeat: Multiply the new value (-2) by 'c' (-2), which is 4. Place it under the next coefficient (-7) and add. -7 + 4 = -3. -2 | 1 0 -7 -6 | -2 4 |____________ 1 -2 -3
  4. Final Step: Multiply the new value (-3) by 'c' (-2), which is 6. Place it under the final coefficient (-6) and add. -6 + 6 = 0. -2 | 1 0 -7 -6 | -2 4 6 |____________ 1 -2 -3 | 0
Conclusion:

The last number (0) is the remainder. The other numbers (1, -2, -3) are the coefficients of the quotient. Since the original polynomial was degree 3, the quotient is degree 2.

  • Quotient Q(x): x² - 2x - 3
  • Remainder R: 0