Synthetic Division

Learn a faster method for dividing polynomials by linear factors using synthetic division.

Advanced25 minLesson
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Definition

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x−c)(x - c).
Instead of using long division, we work only with the coefficients and perform simple arithmetic operations.
When to use synthetic division:
  • The divisor must be linear: (x−c)(x - c) or (x+c)(x + c)
  • The coefficient of xx in the divisor must be 11
The setup: To divide P(x)P(x) by (x−c)(x - c):
  1. 1.Write the value of cc (the root) on the left
  2. 2.Write the coefficients of P(x)P(x) in order
  3. 3.Include 00 for any missing terms
canan−1⋯a0\begin{array}{c|cccc} c & a_n & a_{n-1} & \cdots & a_0 \\ \end{array}

Try it now

When dividing by (x−5)(x - 5), what value of cc do you use in synthetic division?

Worked Examples

Divide (x3−6x2+11x−6)(x^3 - 6x^2 + 11x - 6) by (x−2)(x - 2)
1

Identify the divisor value

(x−2)(x - 2) means c=2c = 2
2

Write coefficients in order

x3−6x2+11x−6x^3 - 6x^2 + 11x - 6 has coefficients: 1,−6,11,−61, -6, 11, -6 → 1−611−61 \quad -6 \quad 11 \quad -6
3

Set up the synthetic division

21−611−6\begin{array}{c|cccc} 2 & 1 & -6 & 11 & -6 \\ & & & & \\ \hline & & & & \end{array}
→ Ready to calculate
4

Bring down the first coefficient

21−611−61\begin{array}{c|cccc} 2 & 1 & -6 & 11 & -6 \\ & & & & \\ \hline & 1 & & & \end{array}
→ First coefficient: 11
5

Multiply and add for each column

2×1=22 \times 1 = 2, then −6+2=−4-6 + 2 = -4 2×(−4)=−82 \times (-4) = -8, then 11+(−8)=311 + (-8) = 3 2×3=62 \times 3 = 6, then −6+6=0-6 + 6 = 0 →
21−611−62−861−430\begin{array}{c|cccc} 2 & 1 & -6 & 11 & -6 \\ & & 2 & -8 & 6 \\ \hline & 1 & -4 & 3 & 0 \end{array}
6

Write the quotient and remainder

Bottom row: 1,−4,31, -4, 3 are quotient coefficients; 00 is remainder → Quotient: x2−4x+3x^2 - 4x + 3, Remainder: 00

Common Mistakes

Forgetting to include zeros for missing terms

Why it's wrong: Every power of xx from highest to constant must have a coefficient. Missing terms have coefficient 00.

Correct: For x4−16x^4 - 16, write: 1,0,0,0,−161, 0, 0, 0, -16 (zeros for x3x^3, x2x^2, and xx)

Using the wrong sign for c when dividing by (x+c)(x + c)

Why it's wrong: Synthetic division uses the root form (x−c)(x - c). When dividing by (x+2)(x + 2), you need c=−2c = -2.

Correct: (x+2)=(x−(−2))(x + 2) = (x - (-2)), so use c=−2c = -2 in the synthetic division setup

Adding when you should multiply, or vice versa

Why it's wrong: The pattern is: multiply by cc, then add to the next coefficient.

Correct: Always: bring down first coefficient, then repeat (multiply by cc, add to next coefficient)

Writing the quotient with the wrong degree

Why it's wrong: The quotient has one degree less than the original polynomial.

Correct: If dividing a cubic (x3x^3) by linear, the quotient is quadratic (x2x^2)

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Synthetic Division of Polynomials: Step-by-Step Examples

by Understand The Math

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Interactive Visual

Polynomial Division

(x3−6x2+11x−6)÷(x−2)\left(x^{3} - 6x^{2} + 11x - 6\right) \div \left(x - 2\right)
2211−6-61111−6-6
x−2x - 2 = 0 gives 22. Write 22 on the left and the coefficients of the dividend in a row.

Function graph

-1012345-6-4-20246xy(1, 0)(2, 0)(3, 0)P(2) = 0

Tap or hover over the graph to read off values.

P(x) = x³ - 6x² + 11x - 6
  • Zero (x-intercept): (1, 0)
  • Zero (x-intercept): (2, 0)
  • Zero (x-intercept): (3, 0)

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Practice Problems

18 problems
Problem 1 of 18
Easy
When dividing by (x−5)(x - 5), what value of cc do you use in synthetic division?

Why It Matters

Synthetic division is a powerful tool that saves time and reduces errors:
  • Speed: Much faster than polynomial long division
  • Simplicity: Only uses basic arithmetic (addition and multiplication)
  • Factoring: Helps find roots of polynomials quickly
  • Remainder Theorem: If you divide P(x)P(x) by (x−c)(x - c), the remainder equals P(c)P(c)
Real applications:
  • Engineering: Analyzing transfer functions in control systems
  • Computer graphics: Polynomial curve calculations
  • Finance: Modeling growth patterns with polynomial functions

Real World Applications

Engineering: Transfer Functions

Engineers use polynomial division when analyzing control systems. The transfer function of a system is often a ratio of polynomials.

Example:

If a system has transfer function x3−6x2+11x−6x−2\frac{x^3 - 6x^2 + 11x - 6}{x - 2}, synthetic division simplifies it to x2−4x+3x^2 - 4x + 3.

1Try It Yourself

A filter circuit has polynomial response P(x)=x3+2x2−5x−6P(x) = x^3 + 2x^2 - 5x - 6. You need to factor it by testing if x=2x = 2 is a root.

Use synthetic division to test if (x−2)(x - 2) is a factor.

Step 1: Write the mathematical expression

Set up: coefficients are 1,2,−5,−61, 2, -5, -6 with c=2c = 2

Computer Science: Algorithm Optimization

Evaluating polynomials efficiently uses ideas from synthetic division. Horner's method is essentially synthetic division for function evaluation.

Example:

To find P(5)P(5) for P(x)=2x3−x2+3x−7P(x) = 2x^3 - x^2 + 3x - 7, use synthetic division with c=5c = 5. The remainder equals P(5)=213P(5) = 213.

2Try It Yourself

You need to evaluate P(x)=x3−4x+2P(x) = x^3 - 4x + 2 at x=3x = 3 for a graphics calculation.

Use synthetic division to find P(3)P(3).

Step 1: Write the mathematical expression

Coefficients: 1,0,−4,21, 0, -4, 2 (remember the missing x2x^2 term!)

Key Takeaways

  • 1Synthetic division is a shortcut for dividing polynomials by (x−c)(x - c)
  • 2Use only the coefficients, including 00 for missing terms
  • 3Pattern: bring down, multiply by cc, add, repeat
  • 4The last number is the remainder; other numbers are quotient coefficients
  • 5For (x+c)(x + c), use c=−cc = -c in the setup
  • 6Remainder Theorem: the remainder equals P(c)P(c)

Frequently Asked Questions

No, synthetic division only works when dividing by a linear binomial (x−c)(x - c) where the coefficient of xx is 11. For other divisors like (2x−3)(2x - 3) or (x2+1)(x^2 + 1), use polynomial long division.
No, synthetic division only works when dividing by a linear binomial (x−c)(x - c) where the coefficient of xx is 11. For other divisors like (2x−3)(2x - 3) or (x2+1)(x^2 + 1), use polynomial long division.
Synthetic division is based on the root form (x−c)(x - c). Since (x+2)=(x−(−2))(x + 2) = (x - (-2)), the value of cc is −2-2. Think of it as finding what makes the divisor equal zero: x+2=0x + 2 = 0 means x=−2x = -2.
When you divide P(x)P(x) by (x−c)(x - c), the remainder you get equals P(c)P(c) - the value of the polynomial at x=cx = c. This means you can use synthetic division to quickly evaluate polynomials!

Glossary

Synthetic division
A shortcut method for dividing a polynomial by a linear binomial (x−c)(x - c) using only coefficients
Divisor
The polynomial you are dividing by; in synthetic division, it must be of the form (x−c)(x - c)
Quotient
The result of the division (excluding the remainder)
Remainder
The amount left over after division; equals P(c)P(c) by the Remainder Theorem
Remainder Theorem
States that when P(x)P(x) is divided by (x−c)(x - c), the remainder equals P(c)P(c)
Factor Theorem
States that (x−c)(x - c) is a factor of P(x)P(x) if and only if P(c)=0P(c) = 0

Formula Card

Synthetic Division Setup

canan−1⋯a0\begin{array}{c|ccc} c & a_n & a_{n-1} & \cdots & a_0 \end{array}

Write $c$ on left, coefficients in descending order on right

The Algorithm

Bring down →\to Multiply by cc →\to Add →\to Repeat

Continue until all coefficients are processed

Result Interpretation

P(x)=(x−c)⋅Q(x)+RP(x) = (x-c) \cdot Q(x) + R

Last number is remainder $R$; others form quotient $Q(x)$

Remainder Theorem

P(c)=RP(c) = R

The remainder equals the polynomial evaluated at $c$

Today's challengeOne problem, three tries

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