Factor The Expression Completely Over The Complex Numbers.x3+5x2+9x+45

Factoring the Expression Completely Over the Complex Numbers: x³ + 5x² + 9x + 45

This article will delve into the process of completely factoring the cubic polynomial expression x³ + 5x² + 9x + 45 over the complex numbers. We'll explore various methods, including the Rational Root Theorem, polynomial long division, and the quadratic formula, to achieve a complete factorization. Understanding this process is crucial for advanced algebra, calculus, and other mathematical fields.

Understanding the Problem

Our goal is to express the polynomial x³ + 5x² + 9x + 45 as a product of linear factors. These factors will have the form (x - r), where 'r' represents the roots (or zeros) of the polynomial. Crucially, we're working over the complex numbers, meaning we'll consider both real and imaginary roots. The Fundamental Theorem of Algebra guarantees that a cubic polynomial will have three roots (though some may be repeated).

Method 1: The Rational Root Theorem and Polynomial Long Division

The Rational Root Theorem helps us identify potential rational roots of the polynomial. This theorem states that any rational root of the polynomial a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ can be expressed in the form p/q, where 'p' is a factor of the constant term (a₀) and 'q' is a factor of the leading coefficient (a_n).

In our case, the constant term is 45 and the leading coefficient is 1. Therefore, the possible rational roots are the factors of 45: ±1, ±3, ±5, ±9, ±15, ±45.

Let's test these potential roots by substituting them into the polynomial:

• f(1) = 1 + 5 + 9 + 45 = 60 ≠ 0
• f(-1) = -1 + 5 - 9 + 45 = 40 ≠ 0
• f(3) = 27 + 45 + 27 + 45 = 144 ≠ 0
• f(-3) = -27 + 45 - 27 + 45 = 36 ≠ 0
• f(5) = 125 + 125 + 45 + 45 = 340 ≠ 0
• f(-5) = -125 + 125 - 45 + 45 = 0

We found a root! x = -5 is a root of the polynomial. This means (x + 5) is a factor.

Now we use polynomial long division to find the other factor:

             x² + 9
x + 5 | x³ + 5x² + 9x + 45
       - (x³ + 5x²)
              0      + 9x + 45
              -       (9x + 45)
                       0

The result of the long division is x² + 9. Therefore, we can rewrite the polynomial as:

(x + 5)(x² + 9)

Method 2: Grouping

Sometimes, polynomials can be factored by grouping terms. Let's try this approach:

x³ + 5x² + 9x + 45 = x²(x + 5) + 9(x + 5)

Notice that (x + 5) is a common factor:

= (x + 5)(x² + 9)

This method confirms the result we obtained through polynomial long division.

Factoring Over the Complex Numbers

We've factored the polynomial into (x + 5)(x² + 9). The first factor (x + 5) represents a real root, x = -5. However, the second factor (x² + 9) is a quadratic that requires further factoring over the complex numbers.

To factor x² + 9, we set it equal to zero and solve for x:

x² + 9 = 0 x² = -9 x = ±√(-9) x = ±3i

Therefore, the factors are (x - 3i) and (x + 3i).

Complete Factorization

The complete factorization of x³ + 5x² + 9x + 45 over the complex numbers is:

(x + 5)(x - 3i)(x + 3i)

Verifying the Factorization

We can verify our factorization by expanding the expression:

(x + 5)(x - 3i)(x + 3i) = (x + 5)(x² + 9) = x³ + 5x² + 9x + 45

This confirms that our factorization is correct.

Further Applications and Considerations

Understanding complex factorization is fundamental to various mathematical applications:

• Solving higher-order equations: Factoring polynomials is essential to finding the roots of equations beyond quadratic equations.
• Partial fraction decomposition: In calculus, this technique uses factorization to simplify complex rational functions for integration.
• Linear algebra: Eigenvalues and eigenvectors, crucial concepts in linear algebra, often involve factoring characteristic polynomials.
• Signal processing and control systems: Polynomial factorization plays a crucial role in analyzing and designing systems described by differential equations.

Dealing with Repeated Roots

In some cases, polynomials may have repeated roots. For instance, a polynomial might have the factor (x-r)², indicating a root 'r' with multiplicity 2. The process of finding these repeated roots requires additional techniques, such as examining the derivative of the polynomial.

Irreducible Polynomials

Not all polynomials can be factored into linear factors over the real or complex numbers. Some polynomials are irreducible, meaning they cannot be factored further without using irrational or complex coefficients. The determination of irreducibility often involves advanced mathematical concepts.

Conclusion

Factoring the expression x³ + 5x² + 9x + 45 completely over the complex numbers involves a systematic approach that combines the Rational Root Theorem, polynomial long division, or grouping, and understanding how to factor quadratic expressions involving complex roots. The complete factorization is (x + 5)(x - 3i)(x + 3i). This process is fundamental to many advanced mathematical applications and highlights the power of complex numbers in solving polynomial equations. Mastering these techniques is crucial for success in higher-level mathematics and related fields. Remember to always verify your factorization by expanding the factors to ensure you've accurately found the original polynomial.