3.5Polynomial Division Methods
Divide polynomials using long division and synthetic division. Apply the Remainder Theorem (f(k) = remainder) and the Factor Theorem (f(k) = 0 iff (x − k) is a factor).
Polynomial division is the key to factoring higher-degree polynomials and finding all their roots. The Remainder Theorem and Factor Theorem connect division to zeros — a connection used throughout algebra and calculus.
Essential Question
How can we divide one polynomial by another, and what do the quotient and remainder tell us about the factors and zeros of the original polynomial?
Lesson Overview
Dividing polynomials is the algebraic equivalent of long division with integers. Given a dividend f(x) and a divisor d(x), we find a quotient q(x) and remainder r(x) so that f(x) = d(x)·q(x) + r(x), where the degree of r(x) is less than the degree of d(x). Two methods — long division and synthetic division — let us perform this efficiently. The Remainder and Factor Theorems then connect division directly to evaluating and factoring polynomials.
Division Algorithm
f(x) = d(x)·q(x) + r(x)
deg(r) < deg(d) always
Remainder & Factor Theorems
Remainder = f(k)
(x−k) is a factor ⟺ f(k) = 0
Worked Examples
Divide (2x³ + 3x² − 11x − 6) ÷ (x + 3) using polynomial long division.
Step 1 — Divide leading terms: 2x³ ÷ x = 2x². Write 2x² in quotient.
Step 2 — Multiply: 2x²(x + 3) = 2x³ + 6x². Subtract from dividend.
(2x³ + 3x²) − (2x³ + 6x²) = −3x². Bring down −11x.
Step 3 — Divide: −3x² ÷ x = −3x. Write −3x in quotient.
Step 4 — Multiply: −3x(x + 3) = −3x² − 9x. Subtract.
(−3x² − 11x) − (−3x² − 9x) = −2x. Bring down −6.
Step 5 — Divide: −2x ÷ x = −2. Write −2 in quotient.
Step 6 — Multiply: −2(x + 3) = −2x − 6. Subtract: remainder = 0.
Use synthetic division to compute (x⁴ − 2x³ + 0x² + x − 5) ÷ (x − 2).
Divisor is (x − 2), so k = 2. Coefficients: 1, −2, 0, 1, −5.
Bring down 1.
1 × 2 = 2; −2 + 2 = 0.
0 × 2 = 0; 0 + 0 = 0.
0 × 2 = 0; 1 + 0 = 1.
1 × 2 = 2; −5 + 2 = −3.
Bottom row: 1, 0, 0, 1 | −3.
Use the Remainder Theorem to find f(3) for f(x) = x³ − 4x² + 2x + 1.
Remainder Theorem: f(3) = remainder when f(x) is divided by (x − 3).
Synthetic division with k = 3, coefficients: 1, −4, 2, 1.
Bring down 1.
1 × 3 = 3; −4 + 3 = −1.
−1 × 3 = −3; 2 + (−3) = −1.
−1 × 3 = −3; 1 + (−3) = −2.
Remainder = −2.
Show that (x − 2) is a factor of p(x) = x³ − 3x² − 4x + 12, then fully factor p(x).
Factor Theorem: (x − 2) is a factor iff p(2) = 0.
Synthetic division, k = 2, coefficients: 1, −3, −4, 12.
Bring down 1.
1 × 2 = 2; −3 + 2 = −1.
−1 × 2 = −2; −4 + (−2) = −6.
−6 × 2 = −12; 12 + (−12) = 0. ✓ Remainder = 0.
Quotient: x² − x − 6 = (x − 3)(x + 2).
Find all zeros of f(x) = x³ + x² − 4x − 4, given that x = −1 is a zero.
Since f(−1) = 0, (x + 1) is a factor. Use synthetic division with k = −1.
Coefficients: 1, 1, −4, −4.
Bring down 1.
1 × (−1) = −1; 1 + (−1) = 0.
0 × (−1) = 0; −4 + 0 = −4.
−4 × (−1) = 4; −4 + 4 = 0. Remainder = 0.
Quotient: x² + 0x − 4 = x² − 4 = (x − 2)(x + 2).
Guided Practice
Divide (3x³ − 5x² + 2x − 4) ÷ (x − 2) using long division.
Hint: Divide 3x³ by x to get the first quotient term, then multiply and subtract. Repeat until the degree of the remainder is less than 1.
Use synthetic division to find the quotient and remainder for (x⁴ + 3x³ − x + 5) ÷ (x + 2).
Hint: k = −2. Write all four coefficients including 0 for the missing x² term: 1, 3, 0, −1, 5.
Use the Remainder Theorem to evaluate g(−2) for g(x) = 2x³ + x² − 5x + 3.
Hint: Perform synthetic division with k = −2 on coefficients 2, 1, −5, 3. The last number in the bottom row is g(−2).
Determine whether (x + 4) is a factor of h(x) = x³ + 2x² − 7x − 8.
Hint: Use the Factor Theorem: compute h(−4) via synthetic division. If the remainder is 0, then (x + 4) is a factor.
Given that x = 3 is a zero of f(x) = x³ − 6x² + 11x − 6, find all other zeros.
Hint: Divide f(x) by (x − 3) using synthetic division. Then factor the resulting quadratic quotient.
Key Vocabulary
Polynomial Long Division
A method of dividing a polynomial f(x) by a divisor d(x) by repeatedly dividing the leading term, multiplying, and subtracting — mirroring integer long division.
Example: (2x³+3x²−11x−6) ÷ (x+3) = 2x²−3x−2
Synthetic Division
A shortcut for dividing a polynomial by a linear binomial (x − k) using only the coefficients. Write k (not −k) and perform multiply-add steps.
Example: k = 2 | 1 −3 2 → quotient coefficients in bottom row
Remainder Theorem
When a polynomial f(x) is divided by (x − k), the remainder equals f(k). Synthetic division is the fastest way to evaluate f(k).
Example: f(3) = remainder when f(x) ÷ (x−3)
Factor Theorem
(x − k) is a factor of f(x) if and only if f(k) = 0. This connects zeros of a polynomial to its linear factors.
Example: f(2) = 0 ⟺ (x−2) is a factor of f(x)
Check Your Understanding
Interactive Practice — 5 Questions
What is the quotient when (x³ − 7x + 6) is divided by (x − 2)?
Synthetic division can only be used when the divisor is of the form:
If f(x) is divided by (x − 5) and the remainder is 7, what is f(5)?
Which condition confirms that (x − 3) is a factor of p(x)?
Divide (2x³ + 5x² − x − 6) ÷ (x + 2) using synthetic division. What is the remainder?
Independent Practice
Independent Practice
Divide (x³ + 4x² − 3x − 18) ÷ (x + 3) using long division.
Divide (4x³ − 8x² + 3x − 6) ÷ (x − 2) using long division.
Use synthetic division: (x³ − 5x² + 8x − 4) ÷ (x − 1).
Use synthetic division: (2x⁴ − 3x³ + x − 7) ÷ (x − 3). Include the 0 coefficient for x².
Use the Remainder Theorem to find f(4) for f(x) = x³ − 2x² − 5x + 6.
Common Mistakes
Using k = −3 (the sign from the binomial) in synthetic division for (x + 3).
Rewrite (x + 3) as (x − (−3)), so k = −3 is correct here. For (x − 3), k = +3. Always match the zero of the divisor.
Forgetting to include a 0 coefficient for a missing degree term (e.g., x⁴ + x − 5 has no x³ or x² terms).
Write all coefficients including zeros: 1, 0, 0, 1, −5. Missing a placeholder shifts every column and gives wrong results.
Subtracting incorrectly in long division — forgetting to distribute the negative sign.
When subtracting a row, change every sign in that row. Use parentheses: (3x² − 11x) − (3x² + 9x) = −20x.
Confusing the Remainder Theorem with the Factor Theorem — thinking any remainder means (x−k) is a factor.
Only a remainder of 0 means (x−k) is a factor. A nonzero remainder r means f(k) = r, not that k is a zero.
Math Tips
Synthetic division is much faster than long division for linear divisors — use it whenever the divisor is (x − k).
To find all zeros: use the Rational Zero Theorem to guess a zero, confirm with synthetic division (remainder = 0), then factor the quotient.
The degree of the quotient is always one less than the degree of the dividend when dividing by a linear factor.
You can chain synthetic divisions: after finding one zero, apply synthetic division again to the quotient to find the next zero.
Always verify your answer: multiply d(x)·q(x) + r(x) and check it equals f(x).