POLINOMIAL - 4 (Pembagian suku banyak dengan metode Horner dan Horner Kino)

Introduction

In this tutorial, we will explore the division of polynomials using two methods: Horner's Method and Horner Kino. This guide will help you understand how to divide polynomials with a divisor of degree greater than one, which is essential for solving various mathematical problems, including those found in scholastic and entrance exams.

Step 1: Understanding Polynomial Division

Before diving into the methods, familiarize yourself with the following concepts:

• Polynomial: An expression consisting of variables raised to whole number powers and their coefficients (e.g., ( ax^n + bx^{n-1} + ... + z )).
• Degree of a polynomial: The highest power of the variable in the polynomial.
• Divisor: The polynomial you are dividing by.

Key Points

• Ensure your polynomial is arranged in descending order of the variable's degree.
• Identify the divisor polynomial, which should also be arranged in descending order.

Step 2: Applying Horner's Method

Horner's Method simplifies polynomial division and is particularly useful for linear divisors.

Steps to Use Horner's Method

1. Set Up the Polynomial: Write the coefficients of the polynomial you want to divide.
2. Identify the Root: For a divisor of the form ( x - r ), identify the value of ( r ).
3. Perform Synthetic Division:
   • Write ( r ) to the left and the coefficients of the polynomial to the right.
   • Drop the first coefficient directly down.
   • Multiply ( r ) by the coefficient just dropped and add it to the next coefficient.
   • Repeat this process until all coefficients are processed.

Example

For dividing ( 2x^3 + 3x^2 - 4x + 5 ) by ( x - 1 ):

• Coefficients: [2, 3, -4, 5]
• Root ( r = 1 )

Perform the synthetic division like this:

  1 | 2  3  -4  5
    |    2   5  1
    ----------------
      2  5   1  6

The result is ( 2x^2 + 5x + 1 ) with a remainder of 6.

Step 3: Implementing Horner Kino Method

Horner Kino is a variation suitable for higher-degree divisors.

Steps to Use Horner Kino Method

1. Identify the Polynomial and Divisor: The divisor should be of the form ( (x - r)^n ).

2. Create a Table:

   • Write down the coefficients of the dividend polynomial.
   • Prepare to handle multiple roots.

3. Perform the Division:

   • Similar to Horner's Method, but you will repeat the synthetic division for each root.
   • Adjust the coefficients after each division to reflect the new polynomial.

Example

For dividing ( 2x^4 + 3x^3 - 2x^2 + x - 5 ) by ( (x - 1)^2 ):

• Start with the coefficients: [2, 3, -2, 1, -5]
• Follow the synthetic division process twice, once for each root.

Conclusion

In this tutorial, we covered the essential techniques for dividing polynomials using Horner's and Horner Kino methods. Remember to:

• Arrange polynomials in descending order.
• Carefully perform synthetic division, especially with multiple roots.
• Practice with various polynomial examples to gain confidence.

For further learning, explore more advanced polynomial operations, or tackle specific problems from your studies or exams.