How to Do Polynomial Long Division: Step-by-Step Guide

How to Do Polynomial Long Division: Step-by-Step Guide

Polynomial long division is a method for dividing one polynomial by another, just like long division with numbers. It helps you find the quotient and remainder when dividing polynomials. The result follows the formula P(x) = D(x) * Q(x) + R(x), where P(x) is the dividend, D(x) is the divisor, Q(x) is the quotient, and R(x) is the remainder. For more on the formula, see our Polynomial Long Division Formula page. This guide will walk you through the manual steps with examples.

You'll Need:

• A pencil and paper
• Understanding of polynomial terms (coefficients, variables, exponents)
• Basic arithmetic skills

Step-by-Step Instructions

1. Set up the division: Write the dividend (the polynomial being divided) inside the long division symbol and the divisor outside to the left. Write both polynomials in descending order of degree (highest exponent first). If a term is missing (e.g., no x² term), include it with a coefficient of 0 as a placeholder.
2. Divide the first term: Divide the first term of the dividend by the first term of the divisor. Write this result above the division bar as the first term of the quotient.
3. Multiply: Multiply the entire divisor by the new quotient term. Write the product below the dividend, aligning like terms.
4. Subtract: Subtract the product from the dividend. Be careful with signs – subtracting a negative changes to addition.
5. Bring down: Bring down the next term from the dividend (if any) to form a new polynomial.
6. Repeat: Repeat steps 2-5 using the new polynomial as the current dividend until you cannot divide anymore (i.e., the remainder's degree is less than the divisor's degree).
7. Write the result: The expression above the division bar is the quotient, and the final leftover is the remainder. Express the answer as: Quotient + (Remainder / Divisor).

Example 1: Simple Division

Divide (x² + 5x + 6) by (x + 2).

1. Set up: Dividend = x² + 5x + 6, Divisor = x + 2.
2. Divide: x² ÷ x = x. Write x above the bar.
3. Multiply: x * (x + 2) = x² + 2x. Write below.
4. Subtract: (x² + 5x + 6) - (x² + 2x) = 3x + 6.
5. Bring down: No need, we have 3x + 6.
6. Repeat: Divide 3x ÷ x = 3. Write +3 above. Multiply: 3*(x+2)=3x+6. Subtract: (3x+6)-(3x+6)=0.
7. Result: Quotient = x + 3, Remainder = 0. So (x²+5x+6) ÷ (x+2) = x+3.

Example 2: With Remainder and Missing Terms

Divide (2x³ - 5x + 7) by (x² + x). Note there is no x² term in the dividend and no constant in the divisor. Insert placeholders: dividend = 2x³ + 0x² - 5x + 7, divisor = x² + x + 0.

1. Set up: Dividend = 2x³ + 0x² - 5x + 7, Divisor = x² + x.
2. Divide: 2x³ ÷ x² = 2x. Write above.
3. Multiply: 2x * (x² + x) = 2x³ + 2x². Write below.
4. Subtract: (2x³ + 0x² - 5x + 7) - (2x³ + 2x²) = -2x² - 5x + 7.
5. Bring down: Already have the whole expression.
6. Repeat: Divide -2x² ÷ x² = -2. Write -2 above. Multiply: -2*(x²+x) = -2x² -2x. Subtract: (-2x² -5x +7) - (-2x² -2x) = -3x + 7.
7. Check degree: Remainder -3x + 7 has degree 1, divisor degree 2, so stop. Result: Quotient = 2x - 2, Remainder = -3x + 7. Write as: 2x - 2 + (-3x+7)/(x²+x).

Common Pitfalls

• Missing terms: Always include placeholders (coefficient 0) for missing powers to avoid alignment errors. See our guide on Polynomial Long Division with Missing Terms.
• Sign errors: When subtracting, remember that subtracting a negative becomes addition. Double-check your signs.
• Not checking degree: The remainder must have a degree less than the divisor. If not, continue dividing.

For a detailed explanation of how the quotient and remainder relate, check Interpreting Polynomial Long Division Results. If you have more questions, visit our FAQ page. And if you want to save time, use our Polynomial Long Division Calculator to get instant results with step-by-step work.