Polinomial (Bagian 3) - Pembagian Polinomial Cara Bersusun, Horner dan Horner - Kino

Introduction

This tutorial is designed to guide you through the process of polynomial division using three distinct methods: long division, the Horner scheme, and Horner's method. Understanding these techniques is essential for mastering polynomial operations in mathematics, particularly for students in grade 11.

Step 1: Polynomial Long Division

To perform polynomial long division, follow these steps:

1. Set Up the Division: Write the dividend (the polynomial to be divided) under the long division symbol and the divisor (the polynomial you are dividing by) to the left.

2. Divide the First Terms:

   • Divide the leading term of the dividend by the leading term of the divisor.
   • Write the result above the division symbol.

3. Multiply and Subtract:

   • Multiply the entire divisor by the result obtained in the previous step.
   • Subtract this from the dividend.
   • Bring down the next term of the dividend to form a new polynomial.

4. Repeat:

   • Repeat the process of dividing, multiplying, and subtracting until all terms of the dividend are processed.

5. Final Result: The result will consist of a quotient and a remainder. If the degree of the remainder is less than the degree of the divisor, you stop here.

Practical Tips

• Always align like terms when subtracting.
• Check your work by multiplying the quotient by the divisor and adding the remainder to ensure you return to the original polynomial.

Step 2: Using the Horner Scheme

The Horner scheme simplifies polynomial evaluation and division. Here’s how to use it:

1. Write the Polynomial: Write the polynomial in the form ( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 ).

2. Set Up the Division: Use a synthetic division setup for a specific root (for example, ( x - r )).

3. Perform Division:

   • Write the coefficients of the polynomial in a row.
   • Bring down the leading coefficient.
   • Multiply it by the root and add it to the next coefficient.
   • Continue this process across all coefficients.

4. Result: The final row will represent the coefficients of the quotient polynomial, and the last number will be the remainder.

Common Pitfalls

• Be careful with signs when adding and subtracting coefficients.
• Make sure you correctly identify the root for synthetic division.

Step 3: Applying Horner's Method

Horner's method is a compact way to evaluate polynomials directly. Here’s how to apply it:

1. Identify the Polynomial: Start with the polynomial ( P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 ).

2. Evaluate Using Horner’s Method:

   • Rewrite the polynomial as follows: [ P(x) = (((a_n x + a_{n-1}) x + a_{n-2}) x + ... + a_1) x + a_0 ]
   • Start with ( a_n ) and iteratively multiply by ( x ) and add the next coefficient.

3. Calculate the Value: This method allows you to calculate the value of the polynomial at a specific point efficiently.

Real-World Applications

• Horner's method is useful in numerical analysis for evaluating polynomials with high efficiency.
• Used in programming and computer algorithms to minimize computational resources.

Conclusion

In this tutorial, you learned how to perform polynomial division using long division, the Horner scheme, and Horner's method. Mastering these techniques will aid you in solving more complex polynomial problems. For further practice, consider downloading additional exercises and exploring other polynomial-related tutorials.