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Polynomial Division

Divide polynomials using synthetic division or long division. Verify with remainder theorem and factor theorem.

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Why: Understanding polynomial division helps you make better, data-driven decisions.

How: Enter Dividend coefficients (highest degree first), Divisor coefficients to calculate results.

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📐 Examples — Click to Load

e.g. x³-6x²+11x-6
e.g. x-1
poly-div.sh
CALCULATED
Quotient
x^2-5x+6
Remainder
0
Verification
Dividend = Divisor × Quotient: x^3-6x^2+11x-6 = (x-1)(x^2-5x+6) ✓
Factor Theorem
Factor theorem: (x-1) is a factor. P(1) = 0.000000 = 0 ✓
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Quotient Coefficients

Quotient vs Remainder

📐 Calculation Steps

Synthetic division: divisor is (x - 1)
Bring down coefficients, multiply by 1, add.
Quotient coefficients: [1, -5, 6], Remainder: 0

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

📋 Key Takeaways

  • • Long division: Divide leading terms, multiply divisor by quotient term, subtract, repeat.
  • • Synthetic division: Shortcut when divisor is (x−a). Use only coefficients.
  • • Remainder theorem: P(x) ÷ (x−a) has remainder P(a).
  • • Factor theorem: (x−a) is a factor iff P(a)=0.
  • • Verification: Dividend = Divisor × Quotient + Remainder.

💡 Did You Know?

📐Synthetic division works only for divisors of the form (x−a). For other divisors, use long division.Source: Algorithm
🔢The remainder theorem lets you find P(a) without fully dividing—just run synthetic division.Source: Remainder Theorem
⚡If the remainder is 0, the divisor is a factor and the quotient is the other factor.Source: Factor Theorem
📊Polynomial division is used in partial fractions, curve sketching, and finding asymptotes.Source: Applications
🎯The degree of the remainder is always less than the degree of the divisor.Source: Division Algorithm
🧮Roots of the dividend correspond to linear factors (x−r). Division by (x−r) gives the reduced polynomial.Source: Connection to Roots

📖 Long Division Algorithm

1. Arrange both polynomials in descending order. 2. Divide the leading term of the dividend by the leading term of the divisor. 3. Multiply the divisor by this quotient term. 4. Subtract from the dividend. 5. Bring down the next term and repeat until the remainder has degree less than the divisor.

Synthetic Division Shortcut

For (x−a): Write coefficients of dividend. Put a on the left. Bring down first coefficient. Multiply by a, add to next. Repeat. Last value is remainder; rest form quotient.

🎯 Expert Tips

💡 Use Synthetic When Possible

If divisor is (x−a), synthetic division is faster and less error-prone.

💡 Include Zero Coefficients

For x⁴−1, use 1,0,0,0,-1 to preserve degree.

💡 Verify Your Answer

Check: Divisor × Quotient + Remainder = Dividend.

💡 Factor Theorem

Remainder 0 means (x−a) is a factor. Useful for factoring polynomials.

📊 Remainder vs Factor Theorem

TheoremStatement
RemainderP(x) ÷ (x−a) has remainder P(a)
Factor(x−a) is a factor of P(x) iff P(a)=0

❓ FAQ

When can I use synthetic division?

Only when the divisor is a linear binomial of the form (x−a) with leading coefficient 1.

What is the remainder theorem?

When P(x) is divided by (x−a), the remainder equals P(a). So you can find P(a) quickly via synthetic division.

What is the factor theorem?

(x−a) is a factor of P(x) if and only if P(a)=0. So if the remainder is 0, (x−a) is a factor.

How do I enter coefficients?

Comma-separated, highest degree first. E.g., x³−6x²+11x−6 is 1,-6,11,-6.

What if the divisor has degree > 1?

Use long division. Synthetic division only works for linear divisors.

How do I verify my answer?

Multiply divisor × quotient, add remainder. You should get the dividend.

⚠️ Note: Coefficients must be comma-separated, highest degree first. Use 0 for missing terms (e.g., x⁴−1 → 1,0,0,0,-1).

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