Polinomial (Bagian 5) - Cara Menentukan Akar-akar Persamaan Polinomial

Introduction

This tutorial will guide you through the process of determining the roots of polynomial equations by factoring. Based on the instructional video from m4th-lab, you will learn how to identify polynomial equations, understand their roots, and effectively factor them. These skills are essential for high school mathematics and will help you excel in your studies.

Step 1: Understanding Polynomial Equations

• A polynomial equation is a mathematical expression that consists of variables raised to whole number powers and coefficients.

• The general form of a polynomial can be expressed as:

  [ P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 ]

  where ( a_n, a_{n-1}, ..., a_0 ) are constants, and ( n ) is a non-negative integer.

Step 2: Identifying the Roots of Polynomial Equations

• The roots (or zeros) of a polynomial are the values of ( x ) that make the polynomial equal to zero.
• To find the roots, you can use the following methods:

  • Factoring: Expressing the polynomial as a product of simpler polynomials.

  • Using the Quadratic Formula: For second-degree polynomials, use:

    [ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} ]

Step 3: Factoring Polynomial Equations

• To factor a polynomial, follow these steps:
  1. Look for a common factor: If all terms share a common factor, factor it out first.
  2. Identify the form: Check if the polynomial is a difference of squares, perfect square trinomial, or can be factored into linear factors.
  3. Apply factoring techniques:
     • For quadratics: ( ax^2 + bx + c = (px + q)(rx + s) )
     • For higher-degree polynomials, try synthetic division or grouping.

Step 4: Example 1 - Factoring a Polynomial

• Consider the polynomial ( P(x) = x^2 - 5x + 6 ).
• Factor it:

  1. Find two numbers that multiply to ( 6 ) (the constant term) and add to ( -5 ) (the coefficient of ( x )).

  2. The numbers ( -2 ) and ( -3 ) work because:

     [ -2 \times -3 = 6 \quad \text{and} \quad -2 + -3 = -5 ]

  3. Thus, factor the polynomial as:

     [ P(x) = (x - 2)(x - 3) ]

  4. The roots are ( x = 2 ) and ( x = 3 ).

Step 5: Example 2 - Factoring a Higher-Degree Polynomial

• Consider ( P(x) = x^3 - 6x^2 + 11x - 6 ).
• Factor it by grouping:

  1. Group terms: ( (x^3 - 6x^2) + (11x - 6) ).

  2. Factor out common terms in each group:

     [ x^2(x - 6) + 1(11x - 6) ]

  3. Look for further factoring or apply synthetic division if necessary to find roots.

Step 6: Practice Problems

• Try factoring the following polynomials on your own:
  • ( P(x) = x^2 + 7x + 10 )
  • ( P(x) = 2x^2 - 4x - 6 )

Conclusion

In this tutorial, we covered the fundamentals of polynomial equations, how to identify their roots, and the process of factoring them. Practice these skills with the provided examples and problems to enhance your understanding. As you progress, tackle more complex polynomials and explore additional techniques such as synthetic division or the Rational Root Theorem. Happy learning!