Finding Remaining Zeros Using The Conjugate Roots Theorem

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In the realm of polynomial functions, identifying zeros is a fundamental task. Zeros, also known as roots, are the values of x that make the function equal to zero. For polynomials with real coefficients, the Conjugate Roots Theorem offers a powerful tool for finding these zeros. This theorem states that if a complex number a + bi is a zero of a polynomial with real coefficients, then its complex conjugate a - bi is also a zero. Let's delve into how we can apply this theorem to find the remaining zeros of the polynomial function f(x) = x⁴ + 2x³ - 31x² + 58x + 600, given that 4 - 3i is a zero.

Understanding the Conjugate Roots Theorem

The Conjugate Roots Theorem is a cornerstone in dealing with polynomials that have complex roots. Complex roots always come in conjugate pairs when the polynomial has real coefficients. This theorem is incredibly useful because it allows us to immediately identify another zero if we know one complex zero. In our case, we're given that 4 - 3i is a zero of the polynomial f(x) = x⁴ + 2x³ - 31x² + 58x + 600. Since the coefficients of this polynomial are real numbers, we can apply the Conjugate Roots Theorem. The conjugate of 4 - 3i is 4 + 3i. Therefore, 4 + 3i must also be a zero of f(x). This gives us two zeros of the polynomial, which is a great start toward finding all the zeros of this quartic function. Recognizing and applying this theorem efficiently reduces the complexity of the problem, allowing us to proceed with further analysis using simpler algebraic techniques.

Knowing that complex roots occur in conjugate pairs is particularly useful in higher-degree polynomials. For instance, a polynomial of degree four will have four roots (counting multiplicity). If we identify one complex root, we automatically know its conjugate is also a root. This significantly narrows down the possibilities and simplifies the process of finding the remaining roots. The theorem is not just a theoretical concept but a practical tool that streamlines polynomial analysis. In the context of our polynomial f(x) = x⁴ + 2x³ - 31x² + 58x + 600, identifying 4 + 3i as a zero immediately after knowing 4 - 3i reduces the problem from finding four roots to finding the remaining two real roots, which can be done through techniques like polynomial division or synthetic division. Understanding and utilizing the Conjugate Roots Theorem is a fundamental skill in polynomial algebra, essential for solving problems related to polynomial roots and factorization.

The power of the Conjugate Roots Theorem lies in its ability to cut down the workload significantly. Instead of having to search for individual complex roots, the theorem provides a shortcut. If a polynomial has real coefficients, complex roots cannot exist in isolation; they must come in pairs. By recognizing this, we avoid wasting time on fruitless searches and can instead focus on strategic methods like synthetic division or polynomial factorization. Let’s consider an example to illustrate this point. Suppose we were trying to find the roots of a sixth-degree polynomial and discovered one complex root. By the Conjugate Roots Theorem, we immediately know a second root. This leaves us with a fourth-degree polynomial to solve, a much more manageable task than the original sixth-degree polynomial. For the given polynomial f(x) = x⁴ + 2x³ - 31x² + 58x + 600, identifying 4 + 3i as a root after identifying 4 - 3i allows us to form a quadratic factor from these two roots. This quadratic factor can then be divided out of the original polynomial, leaving us with another quadratic to solve, which is significantly easier than solving the original quartic equation directly. Thus, the theorem acts as a strategic enabler, simplifying complex problems into a series of simpler steps, ultimately leading to a more efficient solution.

Utilizing Known Zeros to Find Other Factors

With the knowledge that 4 - 3i and 4 + 3i are zeros of the polynomial, we can construct a quadratic factor. This is achieved by understanding that if r is a zero of a polynomial, then (x - r) is a factor of that polynomial. Therefore, (x - (4 - 3i)) and (x - (4 + 3i)) are factors of f(x). Multiplying these two factors together will give us a quadratic factor with real coefficients, which we can then use to reduce the degree of the original polynomial. This step is crucial because it transforms the problem from finding roots of a quartic polynomial to finding roots of a simpler polynomial after division. The resulting quadratic will represent a significant portion of the original polynomial, allowing us to apply familiar techniques for solving quadratic equations. The process of forming and using this quadratic factor is a key step in solving polynomial equations with complex roots, providing a pathway to unravel the remaining zeros systematically.

To form the quadratic factor, we multiply (x - (4 - 3i)) and (x - (4 + 3i)). Let's simplify this expression step by step. First, we distribute the negative signs: (x - 4 + 3i) and (x - 4 - 3i). Now, we multiply these two factors:

(x - 4 + 3i)(x - 4 - 3i) = x² - 4x - 3ix - 4x + 16 + 12i + 3ix - 12i - 9i².

Notice that the imaginary terms (-3ix and +3ix, +12i and -12i) cancel out. Also, recall that i² = -1. So, we have:

x² - 8x + 16 - 9(-1) = x² - 8x + 16 + 9 = x² - 8x + 25.

Therefore, the quadratic factor is x² - 8x + 25. This quadratic factor represents a significant portion of the original fourth-degree polynomial. By finding this factor, we've effectively captured the information related to the complex roots of the polynomial. Now, we can use this factor to divide the original polynomial, reducing its degree and simplifying the problem of finding the remaining zeros. The process of multiplying the linear factors corresponding to the complex roots and simplifying to get a real quadratic factor is a critical algebraic manipulation. It encapsulates the essence of how complex roots interact within polynomials with real coefficients, and it's a technique that’s frequently employed in advanced algebra and calculus.

The importance of accurately forming this quadratic factor cannot be overstated. A small error in the multiplication or simplification can lead to an incorrect quadratic, which will derail the subsequent steps in finding the remaining zeros. Therefore, meticulous care should be taken in expanding the product and combining like terms. Double-checking the calculations, particularly the imaginary terms and the i² term, is crucial. Once the correct quadratic factor is obtained, it serves as a reliable tool for polynomial division, which is the next step in isolating the remaining roots. In our case, the quadratic factor x² - 8x + 25 encapsulates the complex roots 4 - 3i and 4 + 3i perfectly. This ensures that when we divide the original polynomial by this factor, the quotient will represent the remaining portion of the polynomial, specifically the part that contains the other zeros we're seeking. The accuracy of this quadratic factor is the foundation upon which the rest of the solution is built, making it a critical step in the process.

Polynomial Division to Find the Remaining Zeros

Now that we have the quadratic factor x² - 8x + 25, we can perform polynomial division to find the remaining factor of f(x). We divide f(x) = x⁴ + 2x³ - 31x² + 58x + 600 by x² - 8x + 25. This process will give us another quadratic, which will allow us to find the remaining zeros of the polynomial. Polynomial division is a systematic method for dividing one polynomial by another, similar to long division with numbers. It involves dividing the highest degree terms, multiplying back, subtracting, and bringing down the next term, repeating the process until the degree of the remainder is less than the degree of the divisor. This technique allows us to decompose the original polynomial into factors, making it easier to identify all its zeros. Mastering polynomial division is essential for solving higher-degree polynomial equations and understanding polynomial factorization.

When we perform the polynomial division of x⁴ + 2x³ - 31x² + 58x + 600 by x² - 8x + 25, we follow these steps:

  1. Divide the leading term of the dividend (x⁴) by the leading term of the divisor (x²), which gives us x². This is the first term of the quotient.
  2. Multiply the divisor (x² - 8x + 25) by x², resulting in x⁴ - 8x³ + 25x².
  3. Subtract this result from the dividend: (x⁴ + 2x³ - 31x²) - (x⁴ - 8x³ + 25x²) = 10x³ - 56x².
  4. Bring down the next term from the dividend (+58x), giving us 10x³ - 56x² + 58x.
  5. Divide the leading term of this new dividend (10x³) by the leading term of the divisor (x²), which gives us 10x. This is the next term of the quotient.
  6. Multiply the divisor (x² - 8x + 25) by 10x, resulting in 10x³ - 80x² + 250x.
  7. Subtract this result from the current dividend: (10x³ - 56x² + 58x) - (10x³ - 80x² + 250x) = 24x² - 192x.
  8. Bring down the last term from the dividend (+600), giving us 24x² - 192x + 600.
  9. Divide the leading term of this new dividend (24x²) by the leading term of the divisor (x²), which gives us 24. This is the final term of the quotient.
  10. Multiply the divisor (x² - 8x + 25) by 24, resulting in 24x² - 192x + 600.
  11. Subtract this result from the current dividend: (24x² - 192x + 600) - (24x² - 192x + 600) = 0.

The quotient is x² + 10x + 24, and the remainder is 0. This confirms that x² - 8x + 25 is indeed a factor of f(x), and the other factor is x² + 10x + 24. The result of this polynomial division provides us with the crucial quadratic equation that will yield the remaining zeros of the original polynomial. The process, though lengthy, is systematic and, when executed correctly, leads us closer to a complete solution.

Having performed the polynomial division accurately, we now have the original polynomial f(x) expressed as a product of two quadratic factors: f(x) = (x² - 8x + 25)(x² + 10x + 24). The first factor, x² - 8x + 25, corresponds to the complex conjugate roots we already identified. The second factor, x² + 10x + 24, holds the key to the remaining zeros, which we anticipate to be real numbers. The significance of this step is that we have reduced the problem of finding the zeros of a quartic polynomial to finding the zeros of a quadratic polynomial, a task that is significantly more straightforward. This decomposition is a testament to the power of polynomial division in conjunction with the Conjugate Roots Theorem. It demonstrates how strategic application of algebraic principles can simplify complex problems into manageable components.

Solving the Quadratic Equation for Remaining Zeros

Now we need to find the zeros of the quadratic factor x² + 10x + 24. We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Factoring is often the quickest method if the quadratic can be easily factored. In this case, we are looking for two numbers that multiply to 24 and add up to 10. These numbers are 6 and 4. Therefore, we can factor the quadratic as (x + 6)(x + 4). Setting each factor equal to zero gives us the remaining zeros of the polynomial. This step is the culmination of our efforts, providing the final pieces of the puzzle. Solving the quadratic equation is a fundamental skill in algebra, and the ability to do so efficiently is crucial for solving polynomial equations of higher degrees.

Having factored the quadratic expression x² + 10x + 24 into (x + 6)(x + 4), we can now easily find the zeros by setting each factor equal to zero:

x + 6 = 0 implies x = -6.

x + 4 = 0 implies x = -4.

Thus, the remaining zeros of the polynomial are -6 and -4. These are the real roots of the original quartic polynomial, complementing the complex conjugate pair we found earlier. With this, we have successfully identified all four zeros of the polynomial f(x) = x⁴ + 2x³ - 31x² + 58x + 600. The process of factoring the quadratic and solving for the roots is a direct application of the zero-product property, a core concept in algebra. This property states that if the product of two factors is zero, then at least one of the factors must be zero. By applying this principle, we efficiently transitioned from the factored form of the quadratic to its roots, completing the solution process.

The clarity and precision in finding these remaining zeros are paramount. Each step, from factoring the quadratic to setting each factor to zero, must be executed with care to avoid errors. The zeros, -6 and -4, are concrete values that satisfy the polynomial equation x² + 10x + 24 = 0. These real roots, along with the complex conjugate pair 4 - 3i and 4 + 3i, represent the complete set of solutions for the original fourth-degree polynomial. This comprehensive solution set highlights the interplay between real and complex roots in polynomial equations. Moreover, it underscores the importance of mastering fundamental algebraic techniques such as factoring and the zero-product property. These skills are not just isolated tools but rather integral components of a larger problem-solving framework applicable to a wide range of mathematical challenges. The final zeros provide a satisfying conclusion to the problem, demonstrating the power and elegance of algebraic methods in uncovering the hidden structure of polynomial equations.

Conclusion

In conclusion, given that 4 - 3i is a zero of the polynomial function f(x) = x⁴ + 2x³ - 31x² + 58x + 600, we successfully found the remaining zeros using the Conjugate Roots Theorem and polynomial division. The remaining zeros are 4 + 3i, -6, and -4. This process demonstrates the power of the Conjugate Roots Theorem in simplifying the search for polynomial zeros, especially when complex roots are involved. By understanding and applying this theorem, we can efficiently navigate the complexities of polynomial equations and arrive at accurate solutions. This method not only solves the specific problem at hand but also reinforces fundamental concepts in algebra, such as polynomial division, factoring, and the nature of complex roots. The ability to solve such problems is crucial in various fields, including engineering, physics, and computer science, where polynomial functions are frequently encountered.