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Divide Polynomials Using Synthetic Division: A Comprehensive Guide

Divide Polynomials Using Synthetic Division: A Comprehensive Guide

January 07, 2025 Education

Divide Polynomials Using Synthetic Division: A Comprehensive Guide

Synthetic division is a powerful tool for dividing polynomials. It simplifies the process of polynomial division by focusing on the coefficients and avoiding the complexity of traditional long division. In this guide, we will walk you through the steps to perform synthetic division with two example polynomials. Let's dive in!

Example 1: Dividing -x^4 2x^5 - 3x^2 1 by x - 2

When dividing polynomials using synthetic division, we start by writing the polynomial in standard form and identifying the coefficients. It's essential to ensure the polynomial is in the proper order, with the highest degree coming first.

Step 1: Identify the Coefficients

Given: -x^4 2x^5 - 3x^2 1
Polynomial in standard form: 2x^5 - x^4 ^3 - 3x^2 1
Coefficients: [2, -1, 0, -3, 0, 1]

Step 2: Identify the Root of the Divisor

The divisor is x - 2. To find the root, we set it equal to zero and solve for x: x - 2 0, thus x 2.

Step 3: Set Up Synthetic Division

Write the coefficients and the value of the root below:

2    2    -1    0    -3    0    1
  4    6    12    24
-------------------
  2    3    6    9    24

Step 4: Perform Synthetic Division

Bring down the first coefficient, 2. Multiply 2 by 2 to get 4, and add it to -1 to get 3. Multiply 3 by 2 to get 6, and add it to 0 to get 6. Multiply 6 by 2 to get 12, and add it to -3 to get 9. Multiply 9 by 2 to get 18, and add it to 0 to get 18. Multiply 18 by 2 to get 36, and add it to 1 to get 37.

Quotient: 2x^4 3x^3 6x^2 9x 18

Remainder: 37

Final Result

-x^4 2x^5 - 3x^2 1{x - 2} 2x^4 3x^3 6x^2 9x 18

Example 2: Dividing 2x^3 5x^2 - 4x - 5 by 2x 1

Step 1: Identify the Coefficients

Given: 2x^3 5x^2 - 4x - 5
Coefficients: [2, 5, -4, -5]

Step 2: Identify the Root of the Divisor

The divisor is 2x 1. To find the root, set it equal to zero: 2x 1 0; x -0.5 or x -1/2.

Step 3: Set Up Synthetic Division

Write the coefficients and the value of the root below:

-0.5  2    5    -4    -5
 -1    -2    3
-------------------
  2    4    -6    -2

Step 4: Perform Synthetic Division

Bring down the first coefficient, 2. Multiply 2 by -0.5 to get -1, and add it to 5 to get 4. Multiply 4 by -0.5 to get -2, and add it to -4 to get -6. Multiply -6 by -0.5 to get 3, and add it to -5 to get -2.

Quotient: 2x^2 4x - 6

Remainder: -2

Final Result

2x^3 5x^2 - 4x - 5{2x 1} 2x^2 4x - 6 -

Summary of Results

-x^4 2x^5 - 3x^2 1{x - 2} 2x^4 3x^3 6x^2 9x 18

2x^3 5x^2 - 4x - 5{2x 1} 2x^2 4x - 6 -

By following these straightforward steps, you can use synthetic division to efficiently divide polynomials. Synthetic division is particularly useful in algebra and calculus, simplifying the process and making it more manageable.

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