What Is The Completely Factored Form Of X4 8x2 9

What is the Completely Factored Form of x⁴ - 8x² + 9?

Factoring polynomials is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding the behavior of functions. While some polynomials factor easily, others require more sophisticated techniques. The polynomial x⁴ - 8x² + 9 presents a challenge that neatly demonstrates several factoring strategies. This article will explore the complete factorization process step-by-step, revealing the underlying mathematical principles and highlighting useful strategies for tackling similar problems.

Understanding the Polynomial

Before we delve into the factoring process, let's examine the given polynomial: x⁴ - 8x² + 9. Notice that this is a quartic polynomial (degree 4), but it's expressed in terms of x², making it resemble a quadratic equation. This observation is key to our initial approach. We can treat it as a quadratic in disguise by using substitution.

The Substitution Technique

Let's introduce a substitution to simplify the polynomial. Let u = x². Substituting this into our original polynomial, we get:

u² - 8u + 9

This is now a standard quadratic trinomial, which is much easier to handle. We can attempt to factor this quadratic expression using various methods.

Factoring the Quadratic

The most common method for factoring quadratics is to find two numbers that add up to the coefficient of the linear term (-8) and multiply to the constant term (9). Unfortunately, there are no two integers that satisfy these conditions. This means that the quadratic u² - 8u + 9 does not factor nicely using integer coefficients. We'll need to resort to the quadratic formula.

Applying the Quadratic Formula

The quadratic formula provides a general solution for any quadratic equation of the form au² + bu + c = 0:

u = [-b ± √(b² - 4ac)] / 2a

In our case, a = 1, b = -8, and c = 9. Substituting these values into the quadratic formula, we get:

u = [8 ± √((-8)² - 4 * 1 * 9)] / 2 * 1

u = [8 ± √(64 - 36)] / 2

u = [8 ± √28] / 2

u = [8 ± 2√7] / 2

u = 4 ± √7

Therefore, the two solutions for u are u₁ = 4 + √7 and u₂ = 4 - √7.

Back-Substitution and Further Factoring

Remember that we substituted u = x². Now we need to back-substitute to express the solutions in terms of x:

x² = 4 + √7 and x² = 4 - √7

This gives us four solutions for x:

• x = ±√(4 + √7)
• x = ±√(4 - √7)

These are the roots of the original quartic polynomial. However, to express the polynomial in its completely factored form, we need to rewrite it using these roots.

Constructing the Factored Form

We can now express the completely factored form of the quartic polynomial using its roots:

(x - √(4 + √7))(x + √(4 + √7))(x - √(4 - √7))(x + √(4 - √7))

This is the completely factored form of x⁴ - 8x² + 9. Note that this form involves irrational numbers (√7). This is perfectly acceptable, and in many cases, it's the most complete and accurate factorization possible.

Alternative Approaches and Considerations

While the quadratic formula and substitution method provided a clear path to the solution, it's important to acknowledge that other approaches exist, albeit potentially more complex. These could involve techniques like:

• Rational Root Theorem: This theorem helps identify potential rational roots of a polynomial. However, in this case, the roots are irrational, making this method less effective.
• Numerical Methods: For polynomials that are difficult to factor analytically, numerical methods (like Newton-Raphson) can approximate the roots. However, these methods don't provide an exact factored form.
• Complex Numbers: While the problem yielded real roots, more complex quartic polynomials might involve complex roots. Factoring these would necessitate the inclusion of imaginary units (i).

Importance of Complete Factorization

Obtaining the completely factored form is crucial for several reasons:

• Solving Equations: Setting the factored polynomial equal to zero allows for easily finding the roots (solutions) of the corresponding equation x⁴ - 8x² + 9 = 0.
• Analyzing Function Behavior: The factored form reveals the x-intercepts of the function y = x⁴ - 8x² + 9, providing insights into its graph and behavior.
• Simplifying Expressions: In more complex algebraic expressions, complete factorization can simplify calculations and reveal underlying relationships.

Conclusion: A Detailed Exploration

This article provides a detailed, step-by-step walkthrough of factoring the quartic polynomial x⁴ - 8x² + 9. We utilized a substitution technique to convert the quartic into a simpler quadratic, employed the quadratic formula to find the roots, and then reconstructed the factored form using these roots. We also briefly touched upon alternative approaches and emphasized the importance of complete factorization in various algebraic contexts. Remember, while some polynomials factor neatly, others require a more nuanced approach, often necessitating the use of the quadratic formula or other advanced techniques. The journey through this factorization illustrates the power and versatility of algebraic methods. Understanding these techniques is essential for anyone seeking a deep understanding of algebra and its applications.