Factoring Polynomials Completely Using Conjugate Roots Theorem

In this comprehensive guide, we will delve into the process of completely factoring a polynomial function, leveraging the Conjugate Roots Theorem where applicable. Our focus will be on the polynomial function $f(x) = x^4 - 22x^3 + 185x^2 - 734x + 1200$, given that $4 - 3i$ is a zero. This detailed exploration will equip you with the knowledge and skills necessary to tackle similar polynomial factorization problems.

Understanding the Problem

We are presented with a fourth-degree polynomial function, $f(x) = x^4 - 22x^3 + 185x^2 - 734x + 1200$. Our primary objective is to factor this polynomial completely. This means expressing it as a product of linear factors (factors of the form $(x - c)$, where $c$ is a constant) and possibly irreducible quadratic factors. A crucial piece of information provided is that $4 - 3i$ is a zero of the polynomial. This is where the Conjugate Roots Theorem comes into play, a powerful tool for handling polynomials with complex roots.

The Conjugate Roots Theorem

The Conjugate Roots Theorem is a cornerstone in the realm of polynomial factorization, particularly when dealing with complex numbers. It states that if a polynomial with real coefficients has a complex number $a + bi$ (where a and b are real numbers, and i is the imaginary unit, $\sqrt{-1}$) as a zero, then its complex conjugate, $a - bi$, is also a zero. This theorem is invaluable because it allows us to immediately identify another zero of the polynomial given a complex zero, simplifying the factorization process.

In our case, we're given that $4 - 3i$ is a zero of $f(x)$. Since the coefficients of $f(x)$ are real numbers (1, -22, 185, -734, and 1200), the Conjugate Roots Theorem directly applies. Therefore, the complex conjugate of $4 - 3i$, which is $4 + 3i$, must also be a zero of $f(x)$. This gives us two zeros right off the bat, significantly narrowing down the possibilities for the remaining factors.

Knowing that $4 - 3i$ and $4 + 3i$ are zeros, we can construct quadratic factors corresponding to these zeros. This is a key step in reducing the fourth-degree polynomial into more manageable quadratic expressions. The beauty of using complex conjugates lies in the fact that when we multiply the corresponding factors, the imaginary terms neatly cancel out, leaving us with a quadratic factor with real coefficients. This is essential for simplifying the polynomial and making further factorization possible.

Constructing Quadratic Factors

Given the zeros $4 - 3i$ and $4 + 3i$, we can construct the corresponding factors as $(x - (4 - 3i))$ and $(x - (4 + 3i))$. To obtain a quadratic factor, we multiply these two factors together:

$(x - (4 - 3i))(x - (4 + 3i)) = (x - 4 + 3i)(x - 4 - 3i)$

This multiplication can be carried out using the distributive property (often referred to as the FOIL method). A helpful approach is to recognize this product as the difference of squares pattern: $(a + b)(a - b) = a^2 - b^2$, where $a = (x - 4)$ and $b = 3i$. Applying this pattern simplifies the calculation:

$(x - 4 + 3i)(x - 4 - 3i) = (x - 4)^2 - (3i)^2$

Expanding $(x - 4)^2$ gives us $x^2 - 8x + 16$, and squaring $3i$ yields $9i^2$. Since $i^2 = -1$, we have $9i^2 = -9$. Substituting these results into the equation, we get:

$x^2 - 8x + 16 - (-9) = x^2 - 8x + 16 + 9 = x^2 - 8x + 25$

Thus, we have derived the quadratic factor $x^2 - 8x + 25$. This quadratic expression has real coefficients and corresponds to the complex conjugate zeros we identified earlier. Now, we can use this quadratic factor to reduce the original fourth-degree polynomial to a quadratic equation, which will allow us to find the remaining zeros and complete the factorization.

Polynomial Division

Now that we have the quadratic factor $x^2 - 8x + 25$, we can use polynomial division to divide the original polynomial, $f(x) = x^4 - 22x^3 + 185x^2 - 734x + 1200$, by this factor. Polynomial division is a fundamental technique for factoring polynomials, allowing us to reduce the degree of the polynomial and find additional factors.

The process involves dividing the dividend ($x^4 - 22x^3 + 185x^2 - 734x + 1200$) by the divisor ($x^2 - 8x + 25$). The quotient we obtain will be another polynomial, and the remainder should be zero (since $x^2 - 8x + 25$ is a factor of $f(x)$). Here's how the polynomial long division unfolds:

 x^2 - 14x + 48
--------------------------------------------
x^2 - 8x + 25 | x^4 - 22x^3 + 185x^2 - 734x + 1200
              - (x^4 -  8x^3 +  25x^2)
              ----------------------------------
                   -14x^3 + 160x^2 - 734x
                   -(-14x^3 + 112x^2 - 350x)
                   ----------------------------------
                            48x^2 - 384x + 1200
                            -(48x^2 - 384x + 1200)
                            ------------------------
                                         0

The quotient obtained from the division is $x^2 - 14x + 48$, and the remainder is 0, confirming that $x^2 - 8x + 25$ is indeed a factor of $f(x)$. This means we can now express the original polynomial as a product of two quadratic factors:

$f(x) = (x^2 - 8x + 25)(x^2 - 14x + 48)$

This step significantly simplifies the factorization problem, as we have reduced a fourth-degree polynomial into the product of two quadratic polynomials. The next step is to factor the remaining quadratic factor, $x^2 - 14x + 48$, to find the remaining zeros of the polynomial.

Factoring the Remaining Quadratic

We are left with the quadratic factor $x^2 - 14x + 48$. To factor this quadratic, we look for two numbers that multiply to 48 and add up to -14. These numbers are -6 and -8. Therefore, we can factor the quadratic as follows:

$x^2 - 14x + 48 = (x - 6)(x - 8)$

This factorization gives us two more zeros of the polynomial: $x = 6$ and $x = 8$. These are real roots, which means they correspond to linear factors. This completes the factorization of the original fourth-degree polynomial.

Complete Factorization

We have now factored the polynomial completely. Combining the quadratic factor corresponding to the complex roots and the linear factors from the real roots, we can express $f(x)$ as:

$f(x) = (x^2 - 8x + 25)(x - 6)(x - 8)$

Alternatively, we can express the complete factorization in terms of all the linear factors, including those corresponding to the complex roots:

$f(x) = (x - (4 - 3i))(x - (4 + 3i))(x - 6)(x - 8)$

This is the complete factorization of the polynomial $f(x) = x^4 - 22x^3 + 185x^2 - 734x + 1200$. We have successfully expressed the polynomial as a product of its linear factors, taking into account both real and complex roots. This comprehensive factorization provides a complete understanding of the polynomial's behavior and its roots.

Summary and Conclusion

In this guide, we have demonstrated a step-by-step process for completely factoring a polynomial function, given a complex zero. We utilized the Conjugate Roots Theorem to identify the conjugate pair of the complex zero, constructed quadratic factors, performed polynomial division, and factored the remaining quadratic to find all zeros of the polynomial. This process underscores the importance of the Conjugate Roots Theorem in simplifying polynomial factorization, particularly when dealing with complex roots.

By understanding and applying these techniques, you can confidently factor a wide range of polynomial functions, gaining a deeper understanding of their structure and behavior. The ability to factor polynomials is a fundamental skill in algebra and calculus, with applications in various fields, including engineering, physics, and computer science. Mastering these techniques will empower you to solve complex problems and analyze mathematical models with greater proficiency.

This detailed guide provides a solid foundation for tackling polynomial factorization problems. Remember to practice these techniques with different examples to solidify your understanding and build your problem-solving skills. With consistent practice, you will become proficient in factoring polynomials and applying the Conjugate Roots Theorem effectively.