Factor The Expression Completely Over The Complex Numbers. Y4+14y2+49

Factoring Expressions Completely Over the Complex Numbers: A Deep Dive into y⁴ + 14y² + 49

Factoring expressions is a fundamental concept in algebra, crucial for solving equations, simplifying expressions, and understanding the behavior of functions. While factoring over real numbers is relatively straightforward, factoring over the complex numbers opens up a wider realm of possibilities. This article will delve into the complete factorization of the expression y⁴ + 14y² + 49 over the complex numbers, exploring the underlying mathematical principles and techniques involved.

Understanding the Problem: y⁴ + 14y² + 49

Our target expression, y⁴ + 14y² + 49, is a polynomial of degree four. Notice that it's not a simple quadratic, but it resembles one if we consider y² as a single variable. This observation is key to our initial approach. We'll treat y² as a variable, say 'x', and refactor the expression. This will allow us to initially factor it using techniques familiar to quadratic expressions. The ultimate goal, however, is to completely factor the expression, including its complex roots if they exist.

Step 1: Factoring as a Quadratic

Let's substitute x = y². Our expression then becomes:

x² + 14x + 49

This is a perfect square trinomial. Recall that a perfect square trinomial is of the form a² + 2ab + b², which factors to (a + b)². In our case, a = x and b = 7. Therefore, the expression factors to:

(x + 7)²

Step 2: Substituting Back and Initial Factoring

Now, let's substitute y² back in for x:

(y² + 7)²

This is a significant step towards our complete factorization, but it's not yet fully factored over the complex numbers. We have a perfect square, but the term within still contains a sum of squares which can be further factored using complex numbers.

Step 3: Factoring the Sum of Squares Using Complex Numbers

The expression (y² + 7) is a sum of squares. Over the real numbers, this expression is irreducible (cannot be factored further). However, over the complex numbers, we can use the difference of squares factorization along with imaginary units. Recall that i² = -1, where 'i' is the imaginary unit.

We can rewrite 7 as -(–7) and –7 as (i√7)². This transforms our expression into a difference of squares:

y² + 7 = y² – (–7) = y² – (i√7)²

Using the difference of squares factorization (a² – b²) = (a + b)(a – b), we get:

(y + i√7)(y – i√7)

Step 4: Complete Factorization Over Complex Numbers

Now we can substitute this factorization back into our previous result:

(y² + 7)² = [(y + i√7)(y – i√7)]²

This simplifies to:

(y + i√7)²(y – i√7)²

This is the complete factorization of y⁴ + 14y² + 49 over the complex numbers. Each factor represents a root of the original polynomial.

Understanding the Roots and Their Multiplicity

Our factorization reveals four roots:

• y = i√7: This root has a multiplicity of 2.
• y = -i√7: This root also has a multiplicity of 2.

The multiplicity of a root indicates how many times that root appears as a factor in the polynomial's factorization. In this case, both complex conjugate roots, i√7 and -i√7, appear twice each. This is consistent with the fact that our original polynomial was of degree four.

Visualizing the Roots in the Complex Plane

The roots of the polynomial can be visualized as points in the complex plane. Both roots lie on the imaginary axis, symmetrically positioned about the origin. This symmetry is a characteristic of polynomials with only even powers of the variable and real coefficients.

Alternative Approaches and Further Exploration

While the method presented above is a clear and efficient way to factor the given expression, it's worth mentioning that other approaches are possible. For instance:

• Using the quadratic formula: Although initially applying it to the substituted quadratic might seem less direct, the same results would be achieved.
• Polynomial long division: If a root was known beforehand (perhaps through inspection or intuition), polynomial long division could have been used to reduce the polynomial's degree.
• Numerical methods: For more complex polynomials, numerical methods could help approximate roots, though this isn't necessary in this specific case.

The Significance of Complex Numbers in Polynomial Factorization

The use of complex numbers drastically expands our ability to factor polynomials completely. Over the real numbers, many polynomials, especially those with even degrees, are irreducible. However, the fundamental theorem of algebra guarantees that a polynomial of degree n has exactly n roots (counting multiplicities) in the complex number field. This means every polynomial can be fully factored into linear factors if we extend our number system to include complex numbers. This complete factorization provides a profound insight into the polynomial's behavior and properties.

Practical Applications and Conclusion

The concept of factoring polynomials, especially over complex numbers, has significant applications across various branches of mathematics, engineering, and computer science:

• Solving differential equations: The roots of characteristic polynomials determine the solutions of linear differential equations.
• Signal processing: Analyzing signals often involves factoring polynomials in the frequency domain.
• Control systems: Stability analysis of control systems relies on the location of the roots of characteristic polynomials in the complex plane.
• Quantum mechanics: Many quantum mechanical systems involve equations that require factorization in the complex field.

In conclusion, completely factoring the expression y⁴ + 14y² + 49 over complex numbers leads us to (y + i√7)²(y – i√7)². This factorization, obtained through a combination of substitution, factoring of a perfect square trinomial, and the use of imaginary units to factor the sum of squares, provides a complete understanding of the polynomial's roots and their multiplicities. This example showcases the crucial role of complex numbers in achieving complete factorization and unlocks deeper insights into the behavior and applications of polynomials. Understanding these concepts provides a solid foundation for tackling more advanced algebraic and analytical problems.