Synthetic Division for Cubics, Quartics and Missing Terms

You missed a term, your coefficient row is the wrong length, and the quotient comes out looking like nonsense. The fix is the same every time: insert a zero for every missing power before you write the coefficient row. Synthetic division handles missing terms and higher-degree polynomials with worked examples for cubics, quartics, and quintics. A checklist catches the errors that cost you points.

Every missing term means a zero coefficient in the synthetic division table. If the polynomial is 3x⁴ + 2x² + 1, the x³ term and the x term are missing. Write the coefficient row as [3, 0, 2, 0, 1]. The zeroes keep each coefficient in its correct column. Without them, the multiply-and-add steps shift by one position and every result after the shift is wrong.

The same rule applies to every degree. A cubic with no x² term? Write 0 for that coefficient. A quintic with no x⁴ term? Write 0. The quotient's degree is always one less than the dividend's degree, and the coefficient row must have exactly (degree + 1) entries. Count them before you start.

Your dividend is degree 3, so the coefficient row has four entries: [3, −4, 2, −1]. Your divisor is x − 2, so c = 2.

1. Bring down the leading coefficient: 3.
2. Multiply 3 × 2 = 6. Add to the next coefficient (−4): 6 + (−4) = 2. Write 2 below the line.
3. Multiply 2 × 2 = 4. Add to the next coefficient (2): 4 + 2 = 6. Write 6.
4. Multiply 6 × 2 = 12. Add to the last coefficient (−1): 12 + (−1) = 11. Write 11.

The bottom row (excluding the last entry) gives the quotient coefficients: [3, 2, 6], so the quotient is 3x² + 2x + 6. The last entry, 11, is the remainder. The Remainder Theorem tells you that P(2) = 11, which matches the remainder. This is the quickest way to check your work.

The failure mode here is arithmetic, not missing terms. Double-check each multiply-and-add step. A single slip changes every value after it.

Your dividend is x⁴ − 3x² + 5x − 2. The x³ term is missing, so the coefficient row needs a zero in that position: [1, 0, −3, 5, −2]. The divisor is x + 1, so c = −1. The sign flip is the most common student error in synthetic division. Dividing by x + 1 means c = −1, not +1.

1. Bring down 1.
2. 1 × (−1) = −1. Add to 0: −1 + 0 = −1.
3. (−1) × (−1) = 1. Add to −3: 1 + (−3) = −2.
4. (−2) × (−1) = 2. Add to 5: 2 + 5 = 7.
5. 7 × (−1) = −7. Add to −2: −7 + (−2) = −9.

The bottom row is [1, −1, −2, 7, −9]. The first four numbers are the quotient coefficients: 1x³ − 1x² − 2x + 7. The last number, −9, is the remainder. The quotient is degree 3, one less than the dividend's degree 4. If your quotient is also degree 4, you forgot to drop the degree.

Check using the Remainder Theorem: P(−1) should equal −9. P(−1) = 1 − 0 − 3 − 5 − 2 = −9. It matches. This check takes ten seconds and catches sign errors and missing zeros.

Dividing x⁵ − 32 by x − 2 is a synthetic division x⁵ case that forces you to write every missing term. The dividend has only two terms: x⁵ and −32. The coefficient row must contain six entries (degree 5 plus one), with zeros for the x⁴, x³, x², and x terms: [1, 0, 0, 0, 0, −32]. The divisor is x − 2, so c = 2.

1. Bring down 1.
2. 1 × 2 = 2. Add to 0: 2.
3. 2 × 2 = 4. Add to 0: 4.
4. 4 × 2 = 8. Add to 0: 8.
5. 8 × 2 = 16. Add to 0: 16.
6. 16 × 2 = 32. Add to −32: 0.

The bottom row is [1, 2, 4, 8, 16, 0]. The first five numbers give the quotient: x⁴ + 2x³ + 4x² + 8x + 16. The remainder is 0, so x − 2 is a factor. By the Factor Theorem, P(2) = 0, meaning 2 is a root. The quotient, x⁴ + 2x³ + 4x² + 8x + 16, is called the depressed polynomial because its degree is one lower than the original. You can divide it again using repeated synthetic division to test other possible roots from the Rational Zero Theorem.

The failure mode here is the long row of zeros. It is easy to rush through the multiply-and-add steps and skip a column. Write every step. Use the Remainder Theorem to confirm the remainder is 0.

Repeated Division Cuts the Degree

Once you have a depressed polynomial, you can apply repeated synthetic division to reduce the degree further. This is the standard method for factoring a cubic or quartic when you know one root. Use synthetic division on the quotient with a new c value. The process is identical: write the new coefficient row, bring down, multiply, add.

For the quintic example above, the depressed polynomial is x⁴ + 2x³ + 4x² + 8x + 16. If you suspect another root, say x = −2, divide this quartic by x + 2 (so c = −2). The coefficient row is [1, 2, 4, 8, 16]. Run the algorithm. The remainder is 16, not zero, so x = −2 is not a root. Repeat until you reach a quadratic, which you can solve with the quadratic formula.

Each round of repeated synthetic division cuts the polynomial's degree by one. The algorithm never changes. The only thing that grows is the number of steps, and the only thing that can fail is a missing zero or an arithmetic error. Check each remainder with the Remainder Theorem before moving to the next division.

Check the Divisor Form

• Is the divisor in the form x − c? If it is x + c, flip the sign: x + 3 means c = −3. If it is ax − b (e.g., 2x − 3), synthetic division does not work directly. You must first divide by x − b/a and then divide the quotient by a. This extra step is often omitted in textbooks. The source for this rule is OpenStax College Algebra 2e section 5.4.

Count the Terms

• Count the terms in the polynomial. A polynomial of degree n has n + 1 terms if no term is missing. If a term is missing, the coefficient row still needs n + 1 entries. Write a zero for each missing power.

Order and Degree

• Write the coefficient row in descending order of degree. Do not reorder the terms. The order is from highest power to constant term. A term written out of order will produce a quotient that is also out of order.
• Check the quotient's degree. For a linear divisor, the quotient's degree is always one less than the dividend's degree. A cubic gives a quadratic quotient. A quartic gives a cubic quotient. If your quotient has the same degree as the dividend, you misaligned the coefficient row or forgot to drop the degree.

Verify With the Remainder Theorem

• Verify the remainder with the Remainder Theorem. Evaluate P(c) separately. If the remainder from synthetic division does not equal P(c), you made an error. This check catches sign errors, missing zeros, and arithmetic mistakes. OpenStax College Algebra 2e section 5.4 states the theorem: the remainder when dividing by x − c is P(c).