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Synthetic Division Calculator

Divide polynomials quickly using synthetic division

Enter coefficients from highest to lowest degree
For divisor (x - k), enter the value of k

📐Synthetic Division Process

Step 1: Set Up
Write coefficients in a row, write k to the left
Step 2: Bring Down
Bring down the first coefficient
Step 3: Multiply and Add
Multiply by k, add to next coefficient, repeat
Step 4: Result
Bottom row = quotient coefficients + remainder (last value)

💡Why Use Synthetic Division?

Faster
Much quicker than long division
Less Writing
Fewer steps and calculations
Find Roots
Test potential zeros quickly
Remainder Theorem
Remainder = f(k) by theorem

💼Applications

Algebra
• Finding polynomial roots
• Factoring polynomials
• Simplifying expressions
Calculus
• Partial fractions
• Integration
• Limit evaluation
Engineering
• Transfer functions
• System analysis
• Signal processing

⚠Important Notes

• Only works for divisors of form (x - k)
• Must include ALL coefficients (use 0 for missing terms)
• If remainder is 0, then (x - k) is a factor
• Sign of k: dividing by (x - 2) means k = 2, not -2

Synthetic Division Calculator: Divide Polynomials Quickly

Table of Contents - Synthetic Division

How to Use This Calculator - Synthetic Division

Enter the polynomial you want to divide (dividend) and the divisor in the form (x - c). For example, to divide x³ - 6x² + 11x - 6 by (x - 2), enter the coefficients 1, -6, 11, -6 and the value c = 2.

Click "Calculate" to see the synthetic division setup and process. The calculator shows the step-by-step work, displays the quotient and remainder, and verifies the result.

The results show the complete division layout, the quotient polynomial, the remainder value, and confirmation that dividend = (divisor)(quotient) + remainder.

Understanding Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear factor of the form (x - c). It's faster and more compact than long division, using only the coefficients instead of writing out full expressions with variables.

Why it's called "synthetic":

The method synthesizes (builds up) the quotient and remainder through a streamlined process that mimics long division but removes the variables and most of the writing. It's polynomial division with all the unnecessary parts stripped away.

When to use it:

Synthetic division only works when dividing by a linear factor (x - c). If your divisor is x - 3, x + 5, or x, you can use synthetic division. If your divisor is x² + 1 or 2x - 1 with a leading coefficient other than 1, you need regular long division.

The basic setup:

Write the value c (from x - c) outside a division symbol, list all the coefficients of the polynomial in descending power order, bring down the first coefficient, then multiply and add repeatedly until done.

What it gives you:

The bottom row contains the coefficients of the quotient followed by the remainder as the last number. The degree of the quotient is one less than the dividend's degree.

The Remainder Theorem connection:

The remainder when dividing f(x) by (x - c) equals f(c). So synthetic division also evaluates the polynomial at x = c. If the remainder is zero, then c is a root of the polynomial.

Speed advantage:

Synthetic division is much faster than long division once you get the hang of it. For finding roots using the Rational Root Theorem and factoring polynomials, this speed makes a huge difference.

How to Perform Synthetic Division Manually

Let me walk through synthetic division with detailed examples.

Example 1: Basic division

Divide x³ - 6x² + 11x - 6 by (x - 2)

Step 1: Set up

2 |  1  -6  11  -6
  |
  |

The 2 comes from x - 2 = 0, so x = 2. The coefficients are 1, -6, 11, -6.

Step 2: Bring down the first coefficient

2 |  1  -6  11  -6
  |
  |  1

Step 3: Multiply by 2, write under the next coefficient

2 |  1  -6  11  -6
  |      2
  |  1

Step 4: Add the column

2 |  1  -6  11  -6
  |      2
  |  1  -4

Step 5: Repeat - multiply -4 by 2

2 |  1  -6  11  -6
  |      2  -8
  |  1  -4

Step 6: Add

2 |  1  -6  11  -6
  |      2  -8
  |  1  -4   3

Step 7: Repeat - multiply 3 by 2

2 |  1  -6  11  -6
  |      2  -8   6
  |  1  -4   3

Step 8: Add

2 |  1  -6  11  -6
  |      2  -8   6
  |  1  -4   3   0

Step 9: Read the answer Bottom row: 1, -4, 3, 0 Quotient: x² - 4x + 3 Remainder: 0

Since remainder is 0, (x - 2) is a factor! x³ - 6x² + 11x - 6 = (x - 2)(x² - 4x + 3)

Example 2: Non-zero remainder

Divide x³ + 4x² - 5x + 1 by (x + 3)

Step 1: Note that x + 3 = x - (-3), so use c = -3

Step 2: Setup and work through

-3 |  1   4  -5   1
   |     -3  -3  24
   |  1   1  -8  25

Quotient: x² + x - 8 Remainder: 25

Result: (x³ + 4x² - 5x + 1) = (x + 3)(x² + x - 8) + 25

Example 3: Missing term (zero coefficient)

Divide x³ + 2x - 5 by (x - 1)

Step 1: Write with zero for missing x² term Coefficients: 1, 0, 2, -5

1 |  1   0   2  -5
  |      1   1   3
  |  1   1   3  -2

Quotient: x² + x + 3 Remainder: -2

Example 4: Dividing by (x + c) where c is positive

Divide 2x³ - x² + 3 by (x + 2)

Step 1: x + 2 = x - (-2), so c = -2 Coefficients: 2, -1, 0, 3 (zero for missing x term)

-2 |  2  -1   0   3
   |     -4  10 -20
   |  2  -5  10 -17

Quotient: 2x² - 5x + 10 Remainder: -17

Example 5: Using synthetic division to evaluate

Find f(3) for f(x) = x⁴ - 5x² + 7x - 2

Use synthetic division with c = 3:

3 |  1   0  -5   7  -2
  |      3   9  12  57
  |  1   3   4  19  55

The remainder is 55, so f(3) = 55.

Example 6: Finding if a value is a root

Is x = 2 a root of x³ - 3x² + x - 2?

2 |  1  -3   1  -2
  |      2  -2  -2
  |  1  -1  -1  -4

Remainder is -4 (not zero), so x = 2 is NOT a root.

Example 7: Dividing by x (which is x - 0)

Divide x³ + 5x² - 3x + 7 by x

Use c = 0:

0 |  1   5  -3   7
  |      0   0   0
  |  1   5  -3   7

Quotient: x² + 5x - 3 Remainder: 7

(This is equivalent to just removing the constant term for the quotient)

Real-World Applications

Finding polynomial roots:

When using the Rational Root Theorem, synthetic division quickly tests candidates. If the remainder is zero, you've found a root and can factor the polynomial.

Factoring higher-degree polynomials:

After finding one root via synthetic division, the quotient is a lower-degree polynomial. Repeat the process to find all roots and factors.

Evaluating polynomials:

The Remainder Theorem says the remainder equals f(c). Synthetic division is an efficient way to evaluate polynomials at specific values, especially for high degrees.

Simplifying rational expressions:

When simplifying expressions like (x³ - 6x² + 11x - 6)/(x - 2), synthetic division quickly gives the simplified form.

Solving equations:

Polynomial equations can be solved by finding roots. Synthetic division is a key tool in this process.

Cryptography algorithms:

Some encryption methods use polynomial division in finite fields. Synthetic division principles apply in these calculations.

Error correction codes:

Digital communication systems use polynomial division for detecting and correcting transmission errors.

Common Mistakes and How to Avoid Them

Mistake 1: Wrong sign for c

Wrong: For (x + 3), using c = 3

Right: For (x + 3), write it as (x - (-3)), so c = -3. The sign flips.

Why it happens: Not rewriting in (x - c) form first. Always identify c from x - c = 0.

Mistake 2: Forgetting zero coefficients

Wrong: For x³ + 5x - 2, using coefficients 1, 5, -2

Right: Include zero for missing x²: use 1, 0, 5, -2. Every power from highest to zero must be represented.

Why it happens: Only writing visible terms. Synthetic division needs all powers.

Mistake 3: Arithmetic errors

Wrong: Multiplying 3 × -4 = 12

Right: 3 × -4 = -12. Sign errors are common.

Why it happens: Rushing. Double-check multiplication and addition, especially with negatives.

Mistake 4: Misreading the quotient

Wrong: For bottom row 1, -4, 3, 0, writing quotient as x³ - 4x² + 3x

Right: The degree drops by one. It's x² - 4x + 3. The last number is the remainder.

Why it happens: Not remembering the degree decreases. Start quotient with one power less than the dividend.

Mistake 5: Using synthetic division for wrong divisors

Wrong: Trying synthetic division for (x² - 1) or (2x - 3)

Right: Synthetic division only works for (x - c) where the x coefficient is 1. Use long division for other cases.

Why it happens: Thinking synthetic division is universal. It's only for linear factors with leading coefficient 1.

Mistake 6: Adding instead of multiplying

Wrong: Bringing down 1, then adding c instead of multiplying

Right: Multiply the most recent bottom number by c, write it in the next column, then add.

Why it happens: Confusing the steps. It's multiply-down, add-down, repeat.

Mistake 7: Not verifying

Wrong: Accepting the answer without checking

Right: Multiply (divisor)(quotient) + remainder and verify it equals the original dividend.

Why it happens: Overconfidence. Verification catches errors.

Related Topics

How This Calculator Works

Step 1: Parse inputs

Extract polynomial coefficients
Include zeros for missing powers
Identify c from divisor (x - c)

Step 2: Setup array

Create top row with coefficients
Initialize bottom row
Place c outside

Step 3: Bring down first coefficient

Copy first coefficient to bottom row
This starts the process

Step 4: Multiply and add loop

For each remaining coefficient:
  Multiply last bottom number by c
  Write product in middle row
  Add column to get new bottom number
  Repeat

Step 5: Extract results

Bottom row (except last) = quotient coefficients
Last bottom number = remainder
Construct quotient polynomial

Step 6: Verify

Calculate (divisor)(quotient) + remainder
Compare with original dividend
Confirm they match

Step 7: Display

Show synthetic division layout
Display quotient polynomial
Show remainder
Provide verification
Interpret remainder (root if zero)

FAQs

What is synthetic division?

A streamlined method for dividing polynomials by linear factors (x - c), using only coefficients instead of full polynomial long division.

When can I use synthetic division?

Only when dividing by a linear factor of the form (x - c), where the coefficient of x is 1. For other divisors, use polynomial long division.

How do I know what value to use for c?

From (x - c), solve x - c = 0 to get x = c. For (x - 3), use c = 3. For (x + 5), write as (x - (-5)), so c = -5.

What if my polynomial is missing powers?

Include zeros for missing terms. For x³ + 5x - 2, use coefficients 1, 0, 5, -2 (zero for the missing x² term).

What does a zero remainder mean?

Zero remainder means (x - c) is a factor of the polynomial, and c is a root. The polynomial divides evenly.

Can I use this to evaluate polynomials?

Yes! The remainder equals f(c) by the Remainder Theorem. Synthetic division at c evaluates the polynomial.

What's the relationship between quotient and dividend degree?

The quotient's degree is always one less than the dividend's degree. Cubic divided by linear gives quadratic.

How is this different from long division?

Synthetic division is faster and uses less space, but only works for linear divisors (x - c). Long division works for any divisor.

Can the quotient have fractions?

If the original coefficients are integers and you're dividing by (x - c) with integer c, the quotient coefficients will be integers (though the remainder might be non-zero).

What if I make a mistake?

Verify your answer: multiply (x - c) by the quotient, add the remainder, and check if you get the original polynomial. If not, redo the work.

Can I divide by (2x - 4)?

Factor out the 2: it's 2(x - 2). Divide by (x - 2) using synthetic division, then divide the result by 2.

How do I read the bottom row?

All numbers except the last are the quotient's coefficients (starting one degree lower than the dividend). The last number is the remainder.

Can synthetic division be done with decimals or fractions?

Yes, but it's messier. The process is the same, just more careful arithmetic required.

What's the point of this method?

Speed. When finding roots or factoring, you might test dozens of candidates. Synthetic division makes each test quick.

How does this help factor polynomials?

Find a root c (remainder = 0), then use the quotient as your reduced polynomial. Repeat until fully factored.

Can I go backwards?

Yes, multiplication. If quotient is x² + 3x + 2 and remainder is 5 from dividing by (x - 3), the original is (x - 3)(x² + 3x + 2) + 5.

What if my divisor doesn't have a constant?

If dividing by just x, that's x - 0, so c = 0. The synthetic division simplifies significantly.

Why does the remainder equal f(c)?

From the division algorithm: f(x) = (x - c)q(x) + r. Plugging in x = c: f(c) = 0 + r = r.

Do I need to show all the steps?

For homework or tests, yes. Show the setup and each multiply-add step. For personal calculations, experienced users can work faster.

What are the most common errors?

Sign errors (especially with c), forgetting zero coefficients, and misidentifying the quotient degree.