Solving and Factoring the Expression (x^4frac{1}{x^4} - 3)

This article offers a detailed explanation of how to solve and factor the complex expression (x^4frac{1}{x^4} - 3). We will use algebraic techniques to simplify and factorize the given polynomial expression. Understanding these steps is crucial for students learning polynomial factorization and for those dealing with similar complex expressions in advanced mathematics.

Step-by-Step Solution

Let's start with the expression:

(x^4frac{1}{x^4} - 3)

Step 1: Simplify the Expression

The given expression simplifies to:

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

Step 2: Polynomial Form of the Expression

We can rewrite the expression in polynomial form:

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

or equivalently:

(x^4 - 2x 1)

This is a polynomial of degree 4 (quartic), and we will factorize it using algebraic identities.

Step 3: Using Substitution Method

Let us use substitution to make the polynomial easier to handle. Let (y x^2). Then the expression becomes:

(y^2 - 3y 1)

This is a quadratic equation, which we can solve using the quadratic formula:

(y frac{-b pm sqrt{b^2 - 4ac}}{2a})

Here, (a 1), (b -3), and (c 1). Plugging these values in, we get:

(y frac{3 pm sqrt{9 - 4}}{2} frac{3 pm sqrt{5}}{2})

Step 4: Factoring the Quadratic

Using the roots, we can rewrite the quadratic as:

(y^2 - 3y 1 (y - frac{3 sqrt{5}}{2})(y - frac{3 - sqrt{5}}{2}))

Now, substituting back (y x^2) into the factored form, we get:

(x^4 - 3x^2 1 (x^2 - frac{3 sqrt{5}}{2})(x^2 - frac{3 - sqrt{5}}{2}))

Step 5: Simplifying Further

Let's simplify the roots further by taking square roots:

(x^4 - 3x^2 1 left(x - sqrt{frac{3 sqrt{5}}{2}}right)left(x sqrt{frac{3 sqrt{5}}{2}}right)left(x - sqrt{frac{3 - sqrt{5}}{2}}right)left(x sqrt{frac{3 - sqrt{5}}{2}}right))

Conclusion

The expression (x^4frac{1}{x^4} - 3) simplifies to 1 - 3 -2. However, when described as a polynomial, it can be factored as:

(x^4 - 2x^2 1 left(x - sqrt{frac{3 sqrt{5}}{2}}right)left(x sqrt{frac{3 sqrt{5}}{2}}right)left(x - sqrt{frac{3 - sqrt{5}}{2}}right)left(x sqrt{frac{3 - sqrt{5}}{2}}right))

This problem demonstrates the importance of substitution methods and the quadratic formula in factoring and solving complex polynomial expressions.

For further reading and practice, consider exploring more polynomial factorization problems and the general steps for solving quadratic and higher-degree polynomials.