Polinomial Part 2. (Pembagian Cara Bersusun dan Skema Horner ) oleh Tuti Haryani,S.P

Introduction

This tutorial focuses on the division of polynomials using two methods: long division and Horner's method. Learning these techniques is essential for solving polynomial equations and simplifying expressions in algebra. Whether you're a student or anyone interested in mathematics, mastering these methods can enhance your problem-solving skills.

Step 1: Understanding Polynomial Division

Before diving into the methods, ensure you grasp the basics of polynomials.

• A polynomial is an expression made up of variables and coefficients, represented in the form:
  • ( P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 )
• The primary goal of polynomial division is to divide ( P(x) ) by another polynomial ( D(x) ).

Step 2: Long Division of Polynomials

Long division is similar to numerical long division. Follow these steps:

1. Arrange the Polynomials: Write the dividend (the polynomial being divided) and the divisor (the polynomial you are dividing by) in standard form, ensuring all terms are present.

2. Divide the Leading Terms:

   • Take the leading term of the dividend and divide it by the leading term of the divisor.
   • Write the result above the division bar.

3. Multiply and Subtract:

   • Multiply the entire divisor by the result obtained in the previous step.
   • Subtract this product from the dividend to find the new polynomial.

4. Repeat:

   • Repeat the process with the new polynomial (the remainder) until the degree of the remainder is less than the degree of the divisor.

Practical Tips for Long Division

• Always keep your work organized to avoid confusion.
• Check your signs during subtraction; errors often occur here.

Step 3: Using Horner's Method

Horner's method provides a more efficient way to evaluate polynomials and perform division. Here’s how to implement it:

1. Set Up the Polynomial: Write the polynomial in descending order of its powers.

2. Choose a Value: Determine the value of ( x ) at which you want to evaluate the polynomial or the root you are using for division.

3. Create a Synthetic Division Table:

   • Write down the coefficients of the polynomial.
   • Bring down the leading coefficient.

4. Perform Synthetic Division:

   • Multiply the value of ( x ) by the number just brought down.
   • Add the result to the next coefficient.
   • Repeat this process until you reach the last coefficient.

5. Interpret the Result:

   • The last number obtained is the remainder.
   • The coefficients of the new polynomial (from the synthetic division) represent the quotient.

Practical Tips for Horner's Method

• Ensure your coefficients are in the correct order to avoid errors.
• This method is particularly useful for quickly evaluating polynomials at given points.

Conclusion

In this tutorial, we explored two methods for dividing polynomials: long division and Horner's method. Both techniques are fundamental in algebra and provide different advantages depending on the complexity of the polynomials involved.

As a next step, practice both methods with various polynomial examples to strengthen your understanding and proficiency. Consider also exploring polynomial factorization to further enhance your algebra skills.