Factor The Expression Over The Complex Numbers. Y2+49

Factoring the Expression y² + 49 over the Complex Numbers

Factoring quadratic expressions is a fundamental skill in algebra. While factoring expressions like y² - 49 (difference of squares) is straightforward over the real numbers, factoring y² + 49 presents a unique challenge. This expression cannot be factored using only real numbers. However, by venturing into the realm of complex numbers, we can successfully factor this expression. This article will delve deep into the process, exploring the concept of complex numbers, their properties, and how they allow us to factor expressions that are unfactorable within the real number system.

Understanding Complex Numbers

Before we tackle the factorization of y² + 49, it's crucial to understand the concept of complex numbers. Real numbers, which include integers, rational numbers, and irrational numbers, represent points on a number line. Complex numbers extend this concept to a two-dimensional plane called the complex plane. A complex number is represented in the form:

z = a + bi

where:

• 'a' is the real part (a ∈ ℝ)
• 'b' is the imaginary part (b ∈ ℝ)
• 'i' is the imaginary unit, defined as √(-1)

The imaginary unit, 'i', is the key to expanding our number system. It allows us to define the square root of negative numbers, something impossible within the constraints of real numbers.

Properties of Complex Numbers

Complex numbers possess several important properties that are essential for algebraic manipulations, including:

• Addition and Subtraction: Complex numbers are added and subtracted by adding or subtracting their real and imaginary parts separately. For example: (a + bi) + (c + di) = (a + c) + (b + d)i
• Multiplication: Complex numbers are multiplied using the distributive property (FOIL method), remembering that i² = -1. For example: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i
• Division: Dividing complex numbers involves multiplying the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a + bi is a - bi. This eliminates the imaginary part from the denominator.

Factoring y² + 49 using Complex Numbers

Now, let's apply our understanding of complex numbers to factor the expression y² + 49. We can rewrite the expression as:

y² - (-49)

This resembles the difference of squares, but with a negative number under the square root. This is where complex numbers come into play. We can express -49 as (7i)², since (7i)² = 7² * i² = 49 * (-1) = -49.

Therefore, we can rewrite the expression as:

y² - (7i)²

Now, we can apply the difference of squares formula: a² - b² = (a + b)(a - b). In our case, a = y and b = 7i. This gives us:

(y + 7i)(y - 7i)

This is the factored form of y² + 49 over the complex numbers. We have successfully factored the expression by utilizing the properties of complex numbers and the difference of squares formula.

Further Exploration: Complex Conjugates and Polynomials

The concept of complex conjugates plays a significant role in factoring polynomials with complex roots. If a polynomial with real coefficients has a complex root (a + bi), then its complex conjugate (a - bi) is also a root. This property is incredibly useful in determining the complete factorization of polynomials.

For instance, consider a higher-degree polynomial with real coefficients that contains y² + 49 as a factor. The roots of y² + 49, which are 7i and -7i, would appear as conjugate pairs within the complete set of roots for the higher-degree polynomial.

Example of a Higher-Degree Polynomial

Let's consider the polynomial: P(y) = y⁴ + 98y² + 2401. This can be factored as follows:

P(y) = (y² + 49)²

Since we know that y² + 49 = (y + 7i)(y - 7i), we can further expand the factorization:

P(y) = (y + 7i)²(y - 7i)²

This illustrates how the complex conjugate roots (7i and -7i) appear in pairs when dealing with polynomials having real coefficients.

Applications of Complex Numbers

Complex numbers, initially appearing as a mathematical abstraction, have profound applications in various fields:

• Electrical Engineering: Complex numbers are essential in analyzing alternating current (AC) circuits, where they represent impedance and phase.
• Signal Processing: Complex numbers facilitate the representation and manipulation of signals in the frequency domain, using techniques like the Fourier transform.
• Quantum Mechanics: Complex numbers are fundamental to the mathematical framework of quantum mechanics, where wave functions are described using complex numbers.
• Fluid Dynamics: Complex numbers are used in analyzing fluid flow and potential theory.

Conclusion

Factoring y² + 49 over the complex numbers highlights the power and importance of extending the number system beyond real numbers. While this expression remains unfactorable within the real number system, the introduction of complex numbers provides a complete and elegant solution. This process showcases the interconnectedness of mathematical concepts and demonstrates the practical applications of complex numbers in various scientific and engineering disciplines. Mastering the techniques of complex number manipulation is essential for anyone seeking a deeper understanding of algebra and its applications in a wide range of fields. The ability to factor expressions like y² + 49 underscores the richness and utility of the complex number system, expanding our mathematical toolkit and enabling us to solve problems that would otherwise be intractable. Remember, the seemingly simple act of factoring can unlock a world of deeper mathematical understanding and practical applications.