Polinomial (Bagian 4) - Teorema Sisa dan Teorema Faktor

Introduction

This tutorial focuses on the Remainder Theorem and the Factor Theorem, which are essential concepts in polynomial mathematics. Understanding these theorems will enhance your ability to work with polynomials, particularly in solving equations and analyzing functions. This guide is structured to provide clear, step-by-step instructions based on the video content.

Step 1: Understanding the Remainder Theorem

The Remainder Theorem states that when a polynomial ( f(x) ) is divided by ( x - c ), the remainder of this division is equal to ( f(c) ).

How to Apply the Remainder Theorem

1. Identify the polynomial ( f(x) ).
2. Choose a value ( c ) to evaluate.
3. Substitute ( c ) into the polynomial:
   • Calculate ( f(c) ).
4. The result is the remainder.

Example

For ( f(x) = 2x^3 - 3x^2 + 4 ), to find the remainder when divided by ( x - 1 ):

• Calculate ( f(1) = 2(1)^3 - 3(1)^2 + 4 = 2 - 3 + 4 = 3 ).
• The remainder is 3.

Step 2: Additional Applications of the Remainder Theorem

The Remainder Theorem can be extended to multiple examples to solidify understanding.

Example 1

• For ( f(x) = x^2 + 5x + 6 ) and ( c = -2 ):
  • Calculate ( f(-2) = (-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0 ).
  • The remainder is 0.

Example 2

• For ( f(x) = x^3 - 4x + 1 ) and ( c = 2 ):
  • Calculate ( f(2) = (2)^3 - 4(2) + 1 = 8 - 8 + 1 = 1 ).
  • The remainder is 1.

Step 3: Understanding the Factor Theorem

The Factor Theorem states that ( x - c ) is a factor of ( f(x) ) if and only if ( f(c) = 0 ).

How to Use the Factor Theorem

1. Determine if ( f(c) = 0 ) for a chosen ( c ).
2. If true, conclude that ( x - c ) is a factor of the polynomial.

Example

For ( f(x) = x^2 - 3x + 2 ):

• Test ( c = 1 ):
  • ( f(1) = 1^2 - 3(1) + 2 = 0 ).
  • Thus, ( x - 1 ) is a factor.

Step 4: Additional Applications of the Factor Theorem

Explore further examples to deepen your understanding.

Example 1

• For ( f(x) = x^3 - 6x^2 + 11x - 6 ) and ( c = 3 ):
  • Calculate ( f(3) = 3^3 - 6(3^2) + 11(3) - 6 = 0 ).
  • Hence, ( x - 3 ) is a factor.

Example 2

• For ( f(x) = x^2 + 4x + 4 ) and ( c = -2 ):
  • Calculate ( f(-2) = (-2)^2 + 4(-2) + 4 = 0 ).
  • Therefore, ( x + 2 ) is a factor.

Conclusion

In this tutorial, we explored the Remainder Theorem and the Factor Theorem, demonstrating their application through various examples. Mastering these concepts is crucial for further studies in polynomials. For continued learning, consider reviewing the previous parts of the polynomial series or practice with additional polynomial equations.