Complete Factorization of a Fourth Degree Polynomial: x^4 - x^3 - 3x^2 - 4x - 4

Polynomial factorization involves expressing a polynomial as a product of simpler polynomials. This process can be particularly challenging for higher-degree polynomials, such as x^4 - x^3 - 3x^2 - 4x - 4. In this article, we will explore the detailed steps for factoring this fourth-degree polynomial using both synthetic division and Rational Root Theorem.

Understanding the Polynomial

The given polynomial is:

x^4 - x^3 - 3x^2 - 4x - 4

Step-by-Step Factorization

1. Finding Rational Roots with the Rational Root Theorem

To begin the factorization process, we will utilize the Rational Root Theorem. According to this theorem, any rational root of the polynomial x^4 - x^3 - 3x^2 - 4x - 4 is a factor of the constant term, -4, divided by a factor of the leading coefficient, 1. Therefore, the possible rational roots are:

plusmn;1 plusmn;2 plusmn;4

2. Testing Possible Rational Roots

Let's test these roots:

x 1

Substituting x 1 into the polynomial:

(1^4 - 1^3 - 3(1)^2 - 4(1) - 4 1 - 1 - 3 - 4 - 4 -9)

This is not zero, so x 1 is not a root.

x -1

Substituting x -1 into the polynomial:

((-1)^4 - (-1)^3 - 3(-1)^2 - 4(-1) - 4 1 - (-1) - 3 4 - 4 -3)

This is not zero, so x -1 is not a root.

x 2

Substituting x 2 into the polynomial:

(2^4 - 2^3 - 3(2)^2 - 4(2) - 4 16 - 8 - 12 - 8 - 4 0)

x 2 is a root.

3. Using Synthetic Division with x - 2

Using synthetic division by x - 2 to divide x^4 - x^3 - 3x^2 - 4x - 4:

1-1-3-4-4 2 22664 11320

The quotient is x^3 x^2 3x 2.

4. Factoring the Quotient x^3 x^2 3x 2

Next, we factor x^3 x^2 3x 2 using the Rational Root Theorem again:

x -1

Substituting x -1 into x^3 x^2 3x 2:

((-1)^3 (-1)^2 3(-1) 2 -1 1 - 3 2 0)

x -1 is a root.

5. Using Synthetic Division with x 1

Using synthetic division by x 1 to divide x^3 x^2 3x 2:

1132 -1 -1030 1030

The quotient is x^2 1.

6. Final Factorization of x^4 - x^3 - 3x^2 - 4x - 4

Combining all the above steps, the complete factorization is:

(x^4 - x^3 - 3x^2 - 4x - 4 (x - 2)(x 1)(x^2 1))

The quadratic x^2 1 does not factor further over the reals, as its discriminant is negative: (1^2 - 4 cdot 1 cdot 1 -3).

Alternative Method: An Easy Trick

For a more straightforward and quicker method, an old trick based on polynomial division can be utilized. Here's a diagram representing an easier approach to find a particular solution for x^4 - x^3 - 3x^2 - 4x - 4:

The diagram shows that x 2 is a root and provides a visual step-by-step process to find the next equation. The same procedure is followed for the degree 3 equation.

For the degree 3 equation, substituting x -2 into the polynomial yields zero, indicating another root. Proceeding in the same manner for the quadratic equation, we can find its roots using the quadratic formula, yielding the factors:

(x 2, -2, -frac{1 3i}{2}, -frac{1 - 3i}{2})

Conclusion

By following these steps, we have successfully completely factorized the polynomial:

(x^4 - x^3 - 3x^2 - 4x - 4 (x - 2)(x 1)(x^2 1))

This technique can be applied to other polynomials to achieve a more efficient and organized factorization process.