How to Solve: Polynomial Division

How to Solve: Polynomial Division

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Polynomial division is the key to simplifying complex algebra problems—like breaking down a huge pizza into equal slices. Master this, and you’ll crush factoring, rational expressions, and even calculus limits on your exam!

What You Need To Know First

Before diving into polynomial division, you must already understand: 1. Long division with numbers (e.g., 875 ÷ 25) – same logic, but with variables. 2. Polynomial terms and degrees (e.g., x³ is degree 3, x is degree 1). 3. Multiplying polynomials (e.g., (x + 2)(x – 3) = x² – x – 6).

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

1. Polynomial Long Division (MEMORISE THIS)

Formula: Dividend = (Divisor × Quotient) + Remainder

Variables: - Dividend = Polynomial you start with (e.g., x³ + 2x² – 5). - Divisor = Polynomial you divide by (e.g., x – 1). - Quotient = Answer you build step-by-step. - Remainder = What’s left (degree must be less than the divisor’s degree).

When to use: Always works for any polynomial division.

2. Synthetic Division (MEMORISE THIS)

Formula: Only works when the divisor is in the form x – c (e.g., x – 2, x + 3).

Steps: 1. Write c (the number from x – c) in a box. 2. List the coefficients of the dividend (include zeros for missing terms). 3. Bring down the first coefficient. 4. Multiply by c, add to the next coefficient, repeat.

Example: Divide x³ + 4x² – 5x – 6 by x – 2. - c = 2 (from x – 2). - Coefficients: 1, 4, –5, –6. - Synthetic division gives quotient x² + 6x + 7 and remainder 8.

When to use: Faster than long division only if the divisor is x – c.

STEP-BY-STEP METHOD: POLYNOMIAL LONG DIVISION

(Follow these steps exactly. No shortcuts until you master this!)

Step 1: Set Up the Problem

• Write the dividend inside the division bracket.
• Write the divisor outside, to the left.
• Leave space above the bracket for the quotient.

Example: Divide 2x³ + 3x² – 5x + 1 by x – 2.

         ________________
x – 2 ) 2x³ + 3x² – 5x + 1

Step 2: Divide the Leading Terms

• Look at the first term of the dividend (2x³) and the first term of the divisor (x).
• Ask: "What do I multiply x by to get 2x³?"
• Answer: 2x². Write this above the bracket, aligned with x².

         2x²

         ________________
x – 2 ) 2x³ + 3x² – 5x + 1

Step 3: Multiply and Subtract

• Multiply the entire divisor (x – 2) by the term you just wrote (2x²).
• Write the result (2x³ – 4x²) below the dividend, aligned by degree.
• Subtract term by term (change signs and add).

         2x²

         ________________
x – 2 ) 2x³ + 3x² – 5x + 1

       –(2x³ – 4x²)

         ________________

              7x² – 5x

Step 4: Bring Down the Next Term

• Drop the next term (–5x) down to the new line.

         2x²

         ________________
x – 2 ) 2x³ + 3x² – 5x + 1

       –(2x³ – 4x²)

         ________________

              7x² – 5x

Step 5: Repeat the Process

• Divide the new leading term (7x²) by the divisor’s leading term (x).
• Answer: 7x. Write this next to 2x² in the quotient.
• Multiply the divisor by 7x, write below, subtract, and bring down the next term.

         2x² + 7x

         ________________
x – 2 ) 2x³ + 3x² – 5x + 1

       –(2x³ – 4x²)

         ________________

              7x² – 5x

            –(7x² – 14x)

              ________________

                   9x + 1

Step 6: Final Division Step

• Divide 9x by x → 9. Write +9 in the quotient.
• Multiply the divisor by 9, write below, subtract.

         2x² + 7x + 9

         ________________
x – 2 ) 2x³ + 3x² – 5x + 1

       –(2x³ – 4x²)

         ________________

              7x² – 5x

            –(7x² – 14x)

              ________________

                   9x + 1

                 –(9x – 18)

                   ________________

                        19

Step 7: Write the Final Answer

• The quotient is 2x² + 7x + 9.
• The remainder is 19 (degree 0, which is less than the divisor’s degree 1).
• Final answer: 2x² + 7x + 9 + 19/(x – 2).

Why? Because Dividend = (Divisor × Quotient) + Remainder.

Worked Examples

Example 1 – Basic: Divide x³ – 6x² + 11x – 6 by x – 2

Step 1: Set up.

         ________________
x – 2 ) x³ – 6x² + 11x – 6

Step 2: Divide x³ ÷ x = x². Write x² above.

         x²

         ________________
x – 2 ) x³ – 6x² + 11x – 6

Step 3: Multiply x – 2 by x² → x³ – 2x². Subtract.

         x²

         ________________
x – 2 ) x³ – 6x² + 11x – 6

       –(x³ – 2x²)

         ________________

             –4x² + 11x

Step 4: Bring down +11x. Divide –4x² ÷ x = –4x. Write –4x above.

         x² – 4x

         ________________
x – 2 ) x³ – 6x² + 11x – 6

       –(x³ – 2x²)

         ________________

             –4x² + 11x

           –(–4x² + 8x)

             ________________

                   3x – 6

Step 5: Bring down –6. Divide 3x ÷ x = 3. Write +3 above.

         x² – 4x + 3

         ________________
x – 2 ) x³ – 6x² + 11x – 6

       –(x³ – 2x²)

         ________________

             –4x² + 11x

           –(–4x² + 8x)

             ________________

                   3x – 6

                 –(3x – 6)

                   ________________

                        0

Final Answer: x² – 4x + 3 (remainder 0).

What we did and why: - We divided term by term, just like number long division. - The remainder is 0, so x – 2 is a factor of the dividend.

Example 2 – Medium: Divide 3x⁴ + 5x³ – 2x + 1 by x² + 1

Note: Missing x² term! Write it as 0x².

Step 1: Set up.

         ________________
x² + 1 ) 3x⁴ + 5x³ + 0x² – 2x + 1

Step 2: Divide 3x⁴ ÷ x² = 3x². Write 3x² above.

         3x²

         ________________
x² + 1 ) 3x⁴ + 5x³ + 0x² – 2x + 1

Step 3: Multiply x² + 1 by 3x² → 3x⁴ + 3x². Subtract.

         3x²

         ________________
x² + 1 ) 3x⁴ + 5x³ + 0x² – 2x + 1

       –(3x⁴ + 0x³ + 3x²)

         ________________

              5x³ – 3x² – 2x

Step 4: Bring down –2x. Divide 5x³ ÷ x² = 5x. Write +5x above.

         3x² + 5x

         ________________
x² + 1 ) 3x⁴ + 5x³ + 0x² – 2x + 1

       –(3x⁴ + 0x³ + 3x²)

         ________________

              5x³ – 3x² – 2x

            –(5x³ + 0x² + 5x)

              ________________

                   –3x² – 7x + 1

Step 5: Bring down +1. Divide –3x² ÷ x² = –3. Write –3 above.

         3x² + 5x – 3

         ________________
x² + 1 ) 3x⁴ + 5x³ + 0x² – 2x + 1

       –(3x⁴ + 0x³ + 3x²)

         ________________

              5x³ – 3x² – 2x

            –(5x³ + 0x² + 5x)

              ________________

                   –3x² – 7x + 1

                 –(–3x² + 0x – 3)

                   ________________

                         –7x + 4

Final Answer: 3x² + 5x – 3 + (–7x + 4)/(x² + 1).

What we did and why: - We included 0x² to keep terms aligned. - The remainder –7x + 4 has degree 1, which is less than the divisor’s degree (2).

Example 3 – Exam Style: Divide 4x³ – 8x² + 5x – 1 by 2x – 1

Trick: Divisor is 2x – 1, not x – c. Use long division (synthetic won’t work here).

Step 1: Set up.

         ________________
2x – 1 ) 4x³ – 8x² + 5x – 1

Step 2: Divide 4x³ ÷ 2x = 2x². Write 2x² above.

         2x²

         ________________
2x – 1 ) 4x³ – 8x² + 5x – 1

Step 3: Multiply 2x – 1 by 2x² → 4x³ – 2x². Subtract.

         2x²

         ________________
2x – 1 ) 4x³ – 8x² + 5x – 1

       –(4x³ – 2x²)

         ________________

             –6x² + 5x

Step 4: Bring down +5x. Divide –6x² ÷ 2x = –3x. Write –3x above.

         2x² – 3x

         ________________
2x – 1 ) 4x³ – 8x² + 5x – 1

       –(4x³ – 2x²)

         ________________

             –6x² + 5x

           –(–6x² + 3x)

             ________________

                   2x – 1

Step 5: Bring down –1. Divide 2x ÷ 2x = 1. Write +1 above.

         2x² – 3x + 1

         ________________
2x – 1 ) 4x³ – 8x² + 5x – 1

       –(4x³ – 2x²)

         ________________

             –6x² + 5x

           –(–6x² + 3x)

             ________________

                   2x – 1

                 –(2x – 1)

                   ________________

                        0

Final Answer: 2x² – 3x + 1 (remainder 0).

What we did and why: - The divisor had a coefficient other than 1, so we divided carefully. - Remainder is 0, so 2x – 1 is a factor.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. Polynomial division is just like number long division—divide, multiply, subtract, bring down, repeat. Always start by dividing the leading terms. If the divisor is x – c, synthetic division is your shortcut. Watch for missing terms (write 0xⁿ!), and never stop until the remainder’s degree is less than the divisor’s. On exam day, double-check your signs and bring down every term. You’ve got this—now go practice with 3 problems tonight, and you’ll own this topic!