Factor The Expression Completely Over The Complex Numbers. Y4+12y2+36

Factoring the Expression Completely Over the Complex Numbers: y⁴ + 12y² + 36

This article will guide you through the process of completely factoring the expression y⁴ + 12y² + 36 over the complex numbers. We'll explore various factoring techniques, delve into the underlying mathematical principles, and address common questions. Understanding this process is crucial for advanced algebra, calculus, and various engineering applications. The keyword here is complex numbers, meaning we'll consider solutions beyond just real numbers.

Introduction: Understanding the Problem

Our goal is to break down the polynomial expression y⁴ + 12y² + 36 into its simplest factors. This means finding expressions that, when multiplied together, result in the original expression. While seemingly straightforward, factoring over the complex numbers requires a deeper understanding of quadratic equations and the nature of complex roots. The expression itself is a quartic polynomial (degree 4), but we can simplify it by treating it initially as a quadratic in terms of y². This is a key strategy we'll employ throughout this process.

Step 1: Recognizing the Quadratic Form

Notice that the expression y⁴ + 12y² + 36 resembles a quadratic equation. If we let u = y², the expression transforms into:

u² + 12u + 36

This is a much more familiar form. We can now factor this quadratic expression using various methods.

Step 2: Factoring the Quadratic

The quadratic u² + 12u + 36 is a perfect square trinomial. This means it can be factored easily into:

(u + 6)(u + 6) or (u + 6)²

This is a significant simplification.

Step 3: Substituting Back to y

Now, we need to substitute back the original variable, y². Remember that u = y². Therefore, our factored expression becomes:

(y² + 6)(y² + 6) or (y² + 6)²

This is a factorization over the real numbers, but we need to factor completely over the complex numbers.

Step 4: Factoring Over the Complex Numbers

The expression (y² + 6)² only considers real solutions. To find the complex solutions, we need to consider the factors (y² + 6). Notice that y² = -6. This leads us into the realm of complex numbers.

We know that the square root of -1 is denoted by i (the imaginary unit). Therefore, we can rewrite -6 as 6 * -1 or 6i². Taking the square root:

y = ±√(-6) = ±√(6i²) = ±i√6

This gives us two complex roots: y = i√6 and y = -i√6.

To express these roots in the factored form, we write:

(y - i√6)(y + i√6) = y² + 6

Therefore, we can replace each (y² + 6) factor with its complex factorization:

(y² + 6)² = [(y - i√6)(y + i√6)]² = (y - i√6)²(y + i√6)²

This is the complete factorization of the original expression over the complex numbers.

The Complete Factorization

Therefore, the complete factorization of y⁴ + 12y² + 36 over the complex numbers is:

(y - i√6)²(y + i√6)²

This shows all four roots of the quartic polynomial, each with a multiplicity of two.

Mathematical Principles Involved

This factorization problem demonstrates several key mathematical concepts:

Factoring Polynomials: The ability to break down a polynomial into simpler expressions is a fundamental skill in algebra. Different techniques are used for different polynomials (quadratic formula, grouping, etc.).
Perfect Square Trinomials: Recognizing a perfect square trinomial like u² + 12u + 36 simplifies the factoring process considerably.
Quadratic Formula: While not directly used here (because of the perfect square trinomial), the quadratic formula is a vital tool for finding the roots of any quadratic equation, even those with complex roots.
Complex Numbers: The solution requires understanding complex numbers and the imaginary unit i. These are crucial for solving polynomial equations that don't have real roots.
Multiplicity of Roots: Each of the complex roots in our example has a multiplicity of 2, meaning it appears twice in the complete factorization.

Alternative Methods and Considerations

While the method outlined above is efficient for this specific problem (due to the perfect square trinomial), other methods could be employed. For instance:

Using the quadratic formula on y² + 6: Even though we factored easily, applying the quadratic formula to y² + 6 would directly provide the complex roots, y = ±i√6.
Directly factoring the quartic: Though more complex, one could attempt to directly factor the quartic polynomial y⁴ + 12y² + 36 using techniques like grouping or polynomial long division. However, this would be significantly more challenging than our approach.

Frequently Asked Questions (FAQ)

Why is factoring over complex numbers important? Many real-world applications, especially in engineering and physics, involve equations with complex roots. Understanding these roots is crucial for a complete understanding of the system being modeled.
What is the significance of the multiplicity of roots? The multiplicity of a root indicates how many times that root appears as a solution to the polynomial equation. This has implications in graphing the polynomial and analyzing its behavior near the root.
Can all polynomials be factored completely over complex numbers? Yes, the Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots (counting multiplicity). Therefore, any polynomial can be completely factored into linear factors over the complex numbers.
How do I check my factorization? You can verify your factorization by expanding your factored expression. If it returns the original polynomial, your factorization is correct.

Conclusion: A Comprehensive Understanding

Completely factoring the expression y⁴ + 12y² + 36 over the complex numbers reveals a deeper understanding of polynomial factorization, complex numbers, and the Fundamental Theorem of Algebra. By strategically using substitution and then employing our knowledge of complex numbers, we arrived at the complete factorization: (y - i√6)²(y + i√6)². This demonstrates a powerful technique applicable to a broader range of polynomial factoring problems. Remember, the key is to recognize the underlying structure of the polynomial and choose the appropriate factoring method, and to always consider the possibility of complex roots when working with polynomials. This thorough understanding enables effective problem-solving in various mathematical contexts.