Polinomial (Bagian 2) - Menentukan Nilai Polinomial dengan Substitusi dan Skema Horner

Introduction

In this tutorial, we will learn how to evaluate a polynomial using substitution and Horner's method. This technique is crucial for solving polynomial equations effectively, especially for students in advanced mathematics courses. By the end, you will have a clear understanding of how to apply these methods to find the value of polynomials.

Step 1: Understanding Polynomials

Before we dive into evaluation, let's review what a polynomial is:

• A polynomial is a mathematical expression consisting of variables (often represented as x) raised to whole number powers and coefficients.
• An example of a polynomial is ( P(x) = 2x^3 + 3x^2 - x + 5 ).

Key Points

• Polynomials can be of varying degrees based on the highest power of the variable.
• Coefficients are the numerical factors in front of the variables.

Step 2: Substitution Method

To evaluate a polynomial using substitution:

1. Identify the polynomial you want to evaluate.
2. Choose a specific value to substitute for the variable. For example, let’s use ( x = 2 ) for ( P(x) ).
3. Replace the variable in the polynomial with the chosen value.

Example

Using the polynomial ( P(x) = 2x^3 + 3x^2 - x + 5 ):

• Substitute ( x = 2 ):
  • ( P(2) = 2(2^3) + 3(2^2) - 2 + 5 )
  • ( P(2) = 2(8) + 3(4) - 2 + 5 )
  • Calculate each term:
    • ( 2(8) = 16 )
    • ( 3(4) = 12 )
  • Combine the results:
    • ( P(2) = 16 + 12 - 2 + 5 = 31 )

Practical Advice

• Always double-check your arithmetic to avoid small mistakes.
• Use parentheses to ensure proper order of operations.

Step 3: Horner's Method

Horner's method is an efficient way to evaluate polynomials. It reduces the number of multiplications and additions needed.

Steps to Apply Horner's Method

1. Rewrite the polynomial in nested form:

   • For ( P(x) = 2x^3 + 3x^2 - x + 5 ):
   • It can be expressed as:
     • ( P(x) = ((2x + 3)x - 1)x + 5 )

2. Substitute the chosen value step-by-step:

   • Start from the innermost expression and work outward.
   • For ( x = 2 ):

     Step 1: Calculate \( 2(2) + 3 = 7 \)
     Step 2: Calculate \( 7(2) - 1 = 13 \)
     Step 3: Calculate \( 13(2) + 5 = 31 \)
     

Benefits of Horner's Method

• It is faster and requires fewer computations.
• Reduces the risk of error in evaluation.

Conclusion

In this tutorial, you learned how to evaluate polynomials using the substitution method and Horner's method. Both techniques are essential for efficiently solving polynomial equations.

Next Steps

• Practice evaluating different polynomials using both methods to reinforce your understanding.
• Explore more complex polynomials and try applying these methods in various mathematical scenarios.