Long Division Polynomials Worksheet With Answers

Long Division of Polynomials Worksheet with Answers: A Comprehensive Guide

Long division of polynomials is a fundamental concept in algebra, crucial for factoring higher-degree polynomials, solving equations, and understanding more advanced algebraic manipulations. This comprehensive guide provides a detailed explanation of the process, accompanied by a practice worksheet with answers to solidify your understanding. We'll cover various aspects, including the steps involved, common mistakes to avoid, and helpful tips to master this essential skill.

Understanding the Process: A Step-by-Step Guide

Long division of polynomials mirrors the process of long division with numbers. The key is to systematically divide the dividend (the polynomial being divided) by the divisor (the polynomial doing the dividing) to obtain the quotient (the result) and the remainder (the amount left over).

Let's illustrate with an example: Divide (6x³ + 11x² - 31x + 15) by (3x - 2).

Step 1: Setup

Write the problem in the traditional long division format:

3x - 2 | 6x³ + 11x² - 31x + 15

Step 2: Divide the Leading Terms

Divide the leading term of the dividend (6x³) by the leading term of the divisor (3x):

6x³ / 3x = 2x²

Write this result (2x²) above the dividend, aligning it with the corresponding term:

      2x²
3x - 2 | 6x³ + 11x² - 31x + 15

Step 3: Multiply and Subtract

Multiply the quotient term (2x²) by the entire divisor (3x - 2):

2x² * (3x - 2) = 6x³ - 4x²

Subtract this result from the corresponding terms in the dividend:

      2x²
3x - 2 | 6x³ + 11x² - 31x + 15
      - (6x³ - 4x²)
      ----------------
              15x² - 31x

Step 4: Bring Down the Next Term

Bring down the next term from the dividend (-31x):

      2x²
3x - 2 | 6x³ + 11x² - 31x + 15
      - (6x³ - 4x²)
      ----------------
              15x² - 31x

Step 5: Repeat Steps 2-4

Repeat steps 2-4 using the new polynomial (15x² - 31x):

Divide the leading term (15x²) by the leading term of the divisor (3x):

15x² / 3x = 5x

Write 5x above the dividend, aligning it with the corresponding term:

      2x² + 5x
3x - 2 | 6x³ + 11x² - 31x + 15
      - (6x³ - 4x²)
      ----------------
              15x² - 31x
              -(15x² - 10x)
              -------------
                     -21x + 15

Repeat again:

Divide -21x by 3x: -21x / 3x = -7

      2x² + 5x - 7
3x - 2 | 6x³ + 11x² - 31x + 15
      - (6x³ - 4x²)
      ----------------
              15x² - 31x
              -(15x² - 10x)
              -------------
                     -21x + 15
                     -(-21x + 14)
                     -----------
                              1

Step 6: The Remainder

The final result is 1, which is the remainder.

Therefore, (6x³ + 11x² - 31x + 15) divided by (3x - 2) is 2x² + 5x - 7 with a remainder of 1. This can be written as:

2x² + 5x - 7 + 1/(3x - 2)

Common Mistakes to Avoid

• Incorrect Sign Handling: Be meticulous with subtracting the results after multiplication. A misplaced sign can throw off the entire calculation.
• Missing Terms: If the dividend is missing a term (e.g., no x² term), include a placeholder with a coefficient of 0 to maintain proper alignment.
• Division Errors: Carefully perform the division of leading terms. A small error here propagates through the rest of the process.
• Ignoring the Remainder: Remember to include the remainder in your final answer, especially if it's not zero.

Long Division of Polynomials Worksheet

Now, let's put your knowledge to the test with the following problems. Remember to follow the steps outlined above.

Instructions: Perform long division for each problem. Show your work.

Problem 1: (x³ + 5x² + 7x + 3) ÷ (x + 3)

Problem 2: (2x³ - 5x² + 3x + 7) ÷ (x - 2)

Problem 3: (3x⁴ - 4x³ + 2x² - 1) ÷ (x + 1)

Problem 4: (x⁴ + 2x² - x + 5) ÷ (x² - 1) (Note: Divisor is a quadratic)

Problem 5: (4x³ + 12x² - x - 15) ÷ (2x + 3)

Problem 6: (5x³ - 22x² + 19x + 6) ÷ (x - 3)

Problem 7: (x⁴ - 16) ÷ (x - 2) (Difference of Squares consideration)

Problem 8: (2x⁴ + 7x³ - 4x² - 27x - 18) ÷ (x + 3)

Problem 9: (6x³ - x² - 11x + 6) ÷ (2x - 3)

Answers to the Worksheet

Problem 1: x² + 2x + 1

Problem 2: 2x² - x + 1 + 9/(x - 2)

Problem 3: 3x³ - 7x² + 9x - 9 + 8/(x + 1)

Problem 4: x² + 1 + (-x + 6)/(x² - 1) (Note: A remainder with multiple terms)

Problem 5: 2x² + 3x - 5

Problem 6: 5x² - 7x - 2

Problem 7: x³ + 2x² + 4x + 8

Problem 8: 2x³ + x² - 7x - 6

Problem 9: 3x² + 4x - 1 + 3/(2x - 3)

Advanced Techniques and Applications

Mastering long division lays the groundwork for understanding more advanced concepts. For instance:

• Synthetic Division: A shortcut method for dividing polynomials by linear divisors (of the form x - c).
• Remainder Theorem: This theorem states that when a polynomial P(x) is divided by (x - c), the remainder is P(c). This is extremely useful for evaluating polynomial functions.
• Factor Theorem: A corollary of the Remainder Theorem, stating that (x - c) is a factor of P(x) if and only if P(c) = 0. This is critical for finding roots and factoring polynomials.

By consistently practicing long division and understanding these related concepts, you'll build a solid foundation in algebra and prepare for more complex mathematical challenges. Remember, practice is key! Work through additional problems, seek help when needed, and celebrate your progress as you master this vital algebraic skill. The ability to efficiently and accurately divide polynomials will serve you well in your future mathematical endeavors.