When dealing with quadratic polynomials, one of the essential skills is factorization. Let's explore how to factorize the given polynomial x2 - 3x - 1 with a detailed step-by-step approach.

Understanding the Polynomial

The given polynomial is x2 - 3x - 1. This is a second-degree (quadratic) polynomial, meaning it has the general form ax2 bx c where a 1, b -3, and c -1.

Factoring the Polynomial

We will follow a methodical approach to factorize the polynomial. The steps are as follows:

Step 1: Grouping Terms

First, let's rewrite the polynomial as x^2 - 3x - 1. We can observe that the polynomial can be grouped in a way that simplifies the factorization process.

One approach is:

x^2 - 3x - 1 (Grouping the terms to make it easier to factorize) x^2 - 4x x - 1 Now, factor by grouping: x(x - 4) 1(x - 1) Factor out the common terms: (x 1)(x - 1)

Step 2: Using the Quadratic Formula

If the polynomial does not factor nicely, we can use the quadratic formula to find the roots. The quadratic formula is given by:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Substituting a 1, b -3, and c -1 into the formula:

x frac{-(-3) pm sqrt{(-3)^2 - 4(1)(-1)}}{2(1)}

Calculate the discriminant:

x frac{3 pm sqrt{9 4}}{2}

x frac{3 pm sqrt{13}}{2}

So, the roots are:

x frac{3 sqrt{13}}{2} and x frac{3 - sqrt{13}}{2}

Step 3: Factoring Using Roots

Using the roots derived from the quadratic formula, we can write the polynomial as:

x^2 - 3x - 1 (x - frac{3 sqrt{13}}{2})(x - frac{3 - sqrt{13}}{2})

For simplicity, we often express the polynomial as:

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

This can be verified by expanding the right-hand side:

(x - 2)(x 1) x^2 - 2x x - 2 x^2 - x - 2

However, this does not match our original polynomial. Hence, the roots solution is more accurate for factorization using the quadratic formula.

Additional Method: Completing the Square

An alternative method to factorize the polynomial is by completing the square:

x^2 - 3x - 1 x^2 - 3x frac{9}{4} - frac{9}{4} - 1

This can be rewritten as:

x^2 - 3x frac{9}{4} - frac{13}{4} (x - frac{3}{2})^2 - frac{13}{4}

However, this form does not directly give the factorization but helps in understanding the roots.

Conclusion

In conclusion, the factorization of x^2 - 3x - 1 is:

(x - frac{3 sqrt{13}}{2})(x - frac{3 - sqrt{13}}{2})

This demonstrates the power of the quadratic formula in solving and simplifying quadratic polynomials.