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Synthetic Division

Divide P(x) by (x−c) using a compact table. Write c, coefficients. Bring down, multiply by c, add. Last number = remainder = P(c). Factor theorem: (x−c)|P(x) iff P(c)=0.

Concept Fundamentals
x−c, use c
Divisor
R=P(c)
Remainder
P(c)=0 ⇒ (x−c)|P
Factor
Compact table
Ruffini

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Only works for divisor x−c (linear, leading coefficient 1). Remainder theorem: no need to complete — just evaluate P(c). If remainder 0, (x−c) is a factor. Use to factor polynomials.

Key quantities
x−c, use c
Divisor
Key relation
R=P(c)
Remainder
Key relation
P(c)=0 ⇒ (x−c)|P
Factor
Key relation
Compact table
Ruffini
Key relation

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Why: Synthetic division is faster than long division when divisor is x−c. Remainder theorem: R=P(c). Factor theorem: test roots quickly. Essential for polynomial factoring and root finding.

How: Write c in box. Row 1: coefficients (highest degree first). Bring down first. Multiply by c, add to next. Repeat. Last number = remainder. Rest = quotient coefficients.

Only works for divisor x−c (linear, leading coefficient 1).Remainder theorem: no need to complete — just evaluate P(c).
Sources:Khan Academy — Synthetic DivisionWolfram MathWorld — Synthetic Division

Run the calculator when you are ready.

Synthetic DivisionDivide by (x−c)

📐 Examples — Click to Load

P(x) ÷ (x - c)

For (x+2), use c = -2. Include 0 for missing powers.

synthetic.sh
CALCULATED
$ divide --coeffs=[1,-6,11,-6] --c=1
Quotient
1, -5, 6
Remainder
0
(x-c) factor?
Yes
P(c)
0
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Quotient Coefficients

Quotient vs Remainder

📐 Synthetic Division Steps

Polynomial P(x) = x^3 - -6x^2 + 11x - -6
Divisor: (x - 1). Synthetic division uses c = 1
Write coefficients in descending order: [1, -6, 11, -6]
Bring down 1. Then: multiply by 1, add to next coefficient.
1 × 1 + -6 = -5
-5 × 1 + 11 = 6
6 × 1 + -6 = 0
Quotient Q(x) = x^2 - -5x + 6
Remainder R = 0
P(x) = (x - 1) · Q(x) + 0
Remainder is 0 ⇒ (x - 1) is a factor of P(x). Factor theorem: P(1) = 0.

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

x³−6x²+11x−6 ÷ (x−1): remainder 0. So (x−1) is a factor.

— Factor Theorem

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Ruffini's rule: compact form of polynomial division by (x−c).

— History

📋 Key Takeaways

  • • Synthetic division: Fast method to divide P(x) by (x - c) using only coefficients
  • • Ruffini's rule: Same algorithm; named after Paolo Ruffini
  • • Remainder theorem: P(c) = remainder when dividing by (x - c)
  • • Factor theorem: (x - c) is a factor iff P(c) = 0 (remainder = 0)
  • • Use 0 for missing powers (e.g., x³ - 5x + 3 → 1, 0, -5, 3)

📖 Algorithm (Why It Works)

  1. Write coefficients in descending order. Include 0 for missing terms.
  2. Write c from (x - c). For (x + a), use c = -a.
  3. Bring down the first coefficient.
  4. Multiply by c, add to next coefficient. Repeat for each column.
  5. Last number = remainder. Others = quotient coefficients.

Connection to long division: synthetic division is the compact form when dividing by a linear binomial.

🔬 Applications

  • Root finding: Test candidates; if remainder 0, you found a root
  • Polynomial evaluation: P(c) = remainder (remainder theorem)
  • Factoring: Repeated division to find all linear factors
  • Depressed polynomial: After dividing by (x - r), quotient has lower degree

❓ FAQ

Why use -a for (x + a)?

We need the root: (x + a) = 0 when x = -a. Synthetic division uses the root value.

What if the polynomial has missing terms?

Use 0 for missing powers. x³ - 5x + 3 has no x² term, so coefficients are 1, 0, -5, 3.

How does this relate to long division?

Synthetic division is the same process but only with coefficients. It works only for divisors of the form (x - c).

When is (x - c) a factor?

When the remainder is 0. By the factor theorem, P(c) = 0 means (x - c) divides P(x).

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