long division with polynomials worksheet

3 min read 24-11-2024

long division with polynomials worksheet

Meta Description: Master long division with polynomials! This comprehensive guide provides a step-by-step walkthrough, practice problems with solutions, and tips for success. Download your free worksheet now and conquer polynomial division!

Long division with polynomials might seem daunting at first, but with a systematic approach, it becomes manageable. This guide will walk you through the process, providing a worksheet with practice problems and solutions to solidify your understanding. We'll break down the steps, offer helpful tips, and explain common pitfalls to avoid.

Understanding Polynomial Long Division

Polynomial long division is the process of dividing one polynomial by another. It's similar to the long division you learned with numbers, but with variables and exponents involved. The goal is to find the quotient and remainder after dividing.

Key Terminology

Before we dive into the process, let's review some key terms:

Dividend: The polynomial being divided (the one on the inside of the division symbol).
Divisor: The polynomial doing the dividing (the one on the outside of the division symbol).
Quotient: The result of the division.
Remainder: The amount left over after the division.

Steps for Polynomial Long Division

Let's illustrate the process with an example: Divide (3x² + 5x + 2) by (x + 2).

Step 1: Setup

Write the problem in long division format:

x + 2 | 3x² + 5x + 2

Step 2: Divide the Leading Terms

Divide the leading term of the dividend (3x²) by the leading term of the divisor (x): 3x² / x = 3x. Write this above the division bar.

     3x
x + 2 | 3x² + 5x + 2

Step 3: Multiply and Subtract

Multiply the quotient (3x) by the entire divisor (x + 2): 3x(x + 2) = 3x² + 6x. Subtract this result from the dividend.

     3x
x + 2 | 3x² + 5x + 2
      - (3x² + 6x)
      ------------
           -x + 2

Step 4: Bring Down the Next Term

Bring down the next term from the dividend (-x + 2).

Step 5: Repeat Steps 2 and 3

Divide the new leading term (-x) by the leading term of the divisor (x): -x / x = -1. Write this above the division bar.

Multiply the new quotient (-1) by the divisor (x + 2): -1(x + 2) = -x - 2. Subtract this from the remaining dividend.

     3x - 1
x + 2 | 3x² + 5x + 2
      - (3x² + 6x)
      ------------
           -x + 2
         - (-x - 2)
         ---------
               4

Step 6: The Remainder

The result is 4. This is the remainder.

Therefore, (3x² + 5x + 2) / (x + 2) = 3x - 1 with a remainder of 4. This can also be written as 3x - 1 + 4/(x + 2).

Common Mistakes to Avoid

Incorrect Subtraction: Remember to subtract the entire result of the multiplication. Many errors stem from sign mistakes during subtraction.
Missing Terms: If the dividend is missing a term (e.g., no x term), insert a 0x as a placeholder to maintain proper alignment.
Division Errors: Double-check your division of the leading terms at each step.