How to Factor the Polynomial x3 - x2 - x - 1

In this article, we will walk you through each step of factoring the polynomial x3 - x2 - x - 1 using grouping and other methods. By the end of this guide, you'll have a clear understanding of the polynomials and how to factor them systematically.

Method of Grouping

To factor the polynomial x3 - x2 - x - 1, we can use the method of grouping. Here are the steps in detail:

Step 1: Group the Terms

[ x3 - x2 - x - 1 ]

Step 2: Factor Out Common Factors in Each Group

[ (x3 - x2) - (x 1) ] [ x2(x - 1) - 1(x 1) ]

Step 3: Notice the Common Factor

We notice that (x - 1) is a common factor.

Step 4: Factor Further

[ x - 1 times (x2 - x - 1) ]

Step 5: Combine the Factors

The factored form of the polynomial is: [ (x - 1)(x2 - x - 1) ]

Alternative Methods

The polynomial x3 - x2 - x - 1 can also be approached using other methods such as finding its roots. Let's explore how to find the roots and how to factor the polynomial by doing so.

Using Roots and Factoring by Dividing

If x s is a root then x - s is a factor. By observing the polynomial, we find that x 1 is a root.

Hence, we can express the polynomial as:

[ x3 - x2 - x - 1 (x - 1)(x2 - 1) ]

We can further factor x2 - 1 as:

[ x2 - 1 (x - 1)(x 1) ]

Therefore, the polynomial fully factored is:

[ (x - 1)(x - 1)(x 1) (x - 1)2(x 1) ]

The roots are thus x 1, 1, x -1.

Using the Rational Root Theorem

The rational root theorem helps in identifying potential rational roots of a polynomial. For the polynomial x3 - x2 - x - 1, the constant term a_0 and the leading coefficient a_n are both 1. Therefore, the potential roots are simply plusmn;1.

Checking Roots

Let's check if x 1 is a root:

[ 1^3 - 1^2 - 1 - 1 1 - 1 - 1 - 1 0 ]

Since x 1 is a root, we can factor out (x - 1) and divide the polynomial:

[ frac{x^3 - x^2 - x - 1}{x - 1} x^2 - 1 ]

The polynomial can be rewritten as:

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

We can factor x^2 - 1 as:

[ x^2 - 1 (x - 1)(x 1) ]

Therefore, the complete factorization is:

[ (x - 1)(x - 1)(x 1) (x - 1)^2(x 1) ]

Conclusion

In conclusion, factoring polynomials can be approached using various methods such as grouping, finding roots, and the rational root theorem. Understanding these methods will enhance your problem-solving skills in algebra and polynomial manipulation.