TEOREMA FAKTOR - MATEMATIKA PEMINATAN KELAS XI SMA

Introduction

In this tutorial, we will explore the Theorema Faktor, an essential concept in mathematics, particularly for high school students in Indonesia. Understanding this theorem is crucial for mastering polynomial identities and simplifying mathematical expressions. This guide will provide you with clear steps to grasp the theorem and its applications effectively.

Step 1: Understand Polynomials

• Definition: A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.
• Example: A simple polynomial is ( P(x) = 2x^2 + 3x + 5 ).
• Types of Polynomials:
  • Monomial: A single term, e.g., ( 4x^3 ).
  • Binomial: Two terms, e.g., ( 3x + 2 ).
  • Trinomial: Three terms, e.g., ( x^2 + 2x + 1 ).

Step 2: Learn About Theorema Faktor

• Theorema Faktor: This theorem states that if a polynomial ( P(x) ) is divided by ( (x - a) ), the remainder of this division is equal to ( P(a) ).
• Practical Application: This theorem helps in finding the roots of polynomials and simplifying polynomial long division.

Step 3: Apply Theorema Faktor

1. Identify the Polynomial: Choose a polynomial function, for example, ( P(x) = x^3 - 6x^2 + 11x - 6 ).
2. Select a Value for ( a ): Pick a value ( a ) to test, such as ( a = 2 ).
3. Evaluate ( P(a) ):
   • Substitute ( a ) into the polynomial:

   P(2) = 2^3 - 6(2^2) + 11(2) - 6
   

   • Calculate:

   P(2) = 8 - 24 + 22 - 6 = 0
   

4. Interpret the Result: Since ( P(2) = 0 ), it indicates that ( (x - 2) ) is a factor of the polynomial.

Step 4: Factor the Polynomial

• Using Synthetic Division: To factor ( P(x) ) completely:
  1. Divide ( P(x) ) by ( (x - 2) ) using synthetic division.
  2. Write the quotient polynomial.
  3. Factor further if possible.

Conclusion

The Theorema Faktor is a powerful tool in polynomial mathematics that simplifies the process of finding roots and factoring polynomials. By mastering this theorem, you can tackle more complex mathematical problems with confidence. As a next step, practice applying this theorem with various polynomial functions to strengthen your understanding and skills.