Factorizing the Polynomial (x^3 - x^2 - 1 0): Analytical and Numerical Approaches

When dealing with the polynomial equation (x^3 - x^2 - 1 0), one of the first steps is to identify if it can be easily factored using rational roots. However, if it cannot be factored neatly, numerical methods or graphing can help us find the roots more accurately.

The Rational Root Theorem

The Rational Root Theorem suggests that the possible rational roots of the polynomial (x^3 - x^2 - 1) are (pm1), as the constant term is (-1) and the leading coefficient is (1).

Step 1: Testing Rational Roots

Let's test the possible rational roots:

Testing (x 1): ((1)^3 - (1)^2 - 1 1 - 1 - 1 -1) Testing (x -1): ((-1)^3 - (-1)^2 - 1 -1 - 1 - 1 -3)

Since neither (x 1) nor (x -1) are roots, we need to explore other methods to factor the polynomial.

Numerical Methods for Finding Approximate Roots

Through numerical methods, such as the Newton-Raphson method or using graphing calculators, we can find that there is one real root approximately at (x approx 1.3247). We'll denote this root as (r).

Step 2: Synthetic Division

Once we have a root, we can use synthetic division to factor the polynomial. Using the approximate value of (r 1.3247), we can perform synthetic division to get a quadratic polynomial. The result will look like:

(x^3 - x^2 - 1 (x - r)(Ax^2 Bx C))

Step 3: Finding the Quadratic Factor

Using synthetic division, we can determine the coefficients (A), (B), and (C) of the quadratic polynomial (Ax^2 Bx C). This process helps us to rewrite the original polynomial in a factored form.

Step 4: Using the Quadratic Formula

Next, we can solve the quadratic equation (Ax^2 Bx C 0) using the quadratic formula:

(x frac{-B pm sqrt{B^2 - 4AC}}{2A})

This will give us the remaining two roots, which are complex conjugates.

Complex Roots and Their Conjugates

Since the polynomial has only one real root and the other roots are complex, they must be conjugate of each other. If we denote the real root as (r), the complex roots can be written as (u) and (v), where:

(u left(frac{1}{2}left[-frac{29}{27} isqrt{frac{837}{729}}right]right)^{1/3})

(v left(frac{1}{2}left[-frac{29}{27} - isqrt{frac{837}{729}}right]right)^{1/3})

These roots can be used to solve the polynomial (u^3 - u/3 - 29/27 0), which provides the solutions for the polynomial (x^3 - x^2 - 1 0).

Approximate Real Root

After solving for the roots, we find that one of the real roots of the polynomial (x^3 - x^2 - 1 0) is approximately (1.5), which is close to the approximation by Jafar Alaa.

In conclusion, while (x^3 - x^2 - 1 0) may not factor neatly with rational roots, numerical methods and synthetic division can help us find the roots accurately. Understanding the relationship between real and complex roots is crucial for complete factorization.