Factoring the Polynomial (x^4 - 13x^2 36): A Comprehensive Guide

Factoring polynomials can be a challenging task, but it's a fundamental skill in algebra. The polynomial (x^4 - 13x^2 36) is a fourth-degree polynomial, also known as a quartic polynomial. In this article, we will explore the step-by-step process of factoring this polynomial using algebraic techniques.

Overview of the Polynomial

The polynomial in question is (x^4 - 13x^2 36). To factor it, we often use a substitution method, making the process more manageable by reducing it to a quadratic equation.

Using Substitution Method

The first step in factoring the polynomial is to use a substitution. Let (y x^2). This transforms the polynomial into a quadratic equation:

[y^2 - 13y 36 0]

We will now focus on factoring this quadratic equation.

Factoring the Quadratic

To factor the quadratic (y^2 - 13y 36), we need to find two numbers that multiply to 36 and add up to -13. These numbers are -9 and -4. Therefore, we can factor the quadratic as follows:

[y^2 - 13y 36 (y - 9)(y - 4)]

Substituting back (y x^2), we get:

[x^4 - 13x^2 36 (x^2 - 9)(x^2 - 4)]

Further Factoring

Both (x^2 - 9) and (x^2 - 4) are differences of squares, which can be factored using the identity (a^2 - b^2 (a - b)(a b)). Applying this identity, we get:

[x^2 - 9 (x - 3)(x 3)]

[x^2 - 4 (x - 2)(x 2)]

Combining these, we have:

[x^4 - 13x^2 36 (x^2 - 9)(x^2 - 4) (x - 3)(x 3)(x - 2)(x 2)]

Final Factored Form

The fully factored form of the polynomial (x^4 - 13x^2 36) is:

[x^4 - 13x^2 36 (x - 3)(x 3)(x - 2)(x 2)]

Each step is crucial in breaking down the polynomial into its simplest form. Understanding these techniques not only helps in solving similar problems but also enhances one's algebraic skills.

Conclusion

In conclusion, factoring polynomials like (x^4 - 13x^2 36) involves a combination of substitution and the difference of squares identity. By following these systematic steps, we can find the factors of the polynomial, making it easier to solve related algebraic equations and understand the underlying structure of the polynomial.