Factoring polynomial functions completely can seem like a daunting task, especially when dealing with complex roots. However, by leveraging powerful tools like the Conjugate Roots Theorem and employing a systematic approach, we can break down even the most intricate polynomials into manageable factors. This article will guide you through the process, using the example polynomial function f(x) = x⁴ - 7x³ + 14x² - 28x + 40, given that -2i is a zero. Let's embark on this journey of polynomial factorization.
Understanding the Problem
Before diving into the solution, it's crucial to understand the problem statement. We are given a quartic polynomial function, f(x) = x⁴ - 7x³ + 14x² - 28x + 40, and we know that -2i is one of its zeros. Our goal is to factor this polynomial completely, which means expressing it as a product of linear factors (factors of the form x - c, where c is a constant). To achieve this, we will utilize the Conjugate Roots Theorem and other factoring techniques.
The Significance of Zeros
A zero of a polynomial function is a value of x that makes the function equal to zero. In other words, if f(c) = 0, then c is a zero of f(x). Zeros are intimately connected to the factors of a polynomial. If c is a zero, then (x - c) is a factor of the polynomial. This fundamental relationship forms the basis of our factoring strategy.
The Conjugate Roots Theorem: Our Key to Unlocking the Solution
The Conjugate Roots Theorem is a powerful tool that simplifies the process of factoring polynomials with complex coefficients. It states that if a polynomial with real coefficients has a complex number a + bi as a zero, then its complex conjugate a - bi is also a zero. In our case, since the polynomial f(x) has real coefficients and -2i is a zero, the Conjugate Roots Theorem tells us that +2i must also be a zero. This crucial piece of information provides us with two zeros of the polynomial, which will help us construct factors.
Step 1: Applying the Conjugate Roots Theorem
As mentioned earlier, the Conjugate Roots Theorem is pivotal in solving this problem. Since we are given that -2i is a zero of the polynomial f(x) = x⁴ - 7x³ + 14x² - 28x + 40, and the polynomial has real coefficients, we can confidently apply the theorem. The conjugate of -2i is +2i. Therefore, according to the Conjugate Roots Theorem, +2i is also a zero of f(x). Knowing these two complex zeros allows us to construct two quadratic factors.
Constructing Quadratic Factors from Complex Zeros
Now that we know -2i and +2i are zeros of f(x), we can construct corresponding factors. If c is a zero, then (x - c) is a factor. Thus, (x - (-2i)) and (x - 2i) are factors of f(x). These can be simplified to (x + 2i) and (x - 2i), respectively. Multiplying these two factors together will give us a quadratic factor with real coefficients:
(x + 2i)(x - 2i) = x² - (2i)² = x² - (-4) = x² + 4
This quadratic factor, x² + 4, is a significant step forward in our factorization journey. It represents a portion of the original polynomial that we have successfully factored.
Step 2: Polynomial Division to Find Remaining Factors
We've successfully identified a quadratic factor, x² + 4, of the polynomial f(x) = x⁴ - 7x³ + 14x² - 28x + 40. To find the remaining factors, we can perform polynomial long division. We will divide the original polynomial f(x) by the quadratic factor x² + 4. This process will yield another quadratic polynomial, which we can then attempt to factor further.
Performing Polynomial Long Division
Polynomial long division is similar to traditional long division with numbers, but instead of digits, we are working with terms of polynomials. Here's how the division of f(x) by x² + 4 looks:
x² - 7x + 10
x² + 4 | x⁴ - 7x³ + 14x² - 28x + 40
- (x⁴ + 4x²)
------------------
-7x³ + 10x² - 28x
- (-7x³ - 28x)
------------------
10x² + 40
- (10x² + 40)
------------------
0
The result of the division is x² - 7x + 10. The remainder is 0, which confirms that x² + 4 is indeed a factor of f(x).
Interpreting the Result of Division
The quotient we obtained from the long division, x² - 7x + 10, represents the remaining factor of the original polynomial. This means we can now express f(x) as:
f(x) = (x² + 4)(x² - 7x + 10)
We have successfully reduced the problem to factoring a simpler quadratic polynomial.
Step 3: Factoring the Remaining Quadratic
We've arrived at the quadratic expression x² - 7x + 10. Factoring this quadratic will complete the factorization of the original polynomial f(x). There are several techniques to factor a quadratic, such as looking for two numbers that multiply to the constant term and add up to the coefficient of the linear term. In this case, we need two numbers that multiply to 10 and add up to -7.
Finding the Factors of the Quadratic
By careful observation, we can identify that the numbers -2 and -5 satisfy the conditions. (-2) * (-5) = 10 and (-2) + (-5) = -7. Therefore, we can factor the quadratic as follows:
x² - 7x + 10 = (x - 2)(x - 5)
This factorization represents the final piece of the puzzle. We have successfully broken down the quadratic into two linear factors.
Step 4: Complete Factorization
Now that we have factored all the components, we can write the complete factorization of the original polynomial function f(x) = x⁴ - 7x³ + 14x² - 28x + 40. We found the quadratic factor (x² + 4) using the Conjugate Roots Theorem and polynomial division, and we factored the remaining quadratic (x² - 7x + 10) into (x - 2)(x - 5). Combining these results, we get:
f(x) = (x² + 4)(x - 2)(x - 5)
However, we can further factor (x² + 4) using the zeros we found earlier (-2i and 2i):
f(x) = (x + 2i)(x - 2i)(x - 2)(x - 5)
This is the complete factorization of the polynomial f(x) into linear factors. We have successfully expressed the quartic polynomial as a product of four linear factors, two of which have complex coefficients.
Verifying the Complete Factorization
To ensure our factorization is correct, we can expand the factored form and verify that it matches the original polynomial. Expanding the factored form:
(x + 2i)(x - 2i)(x - 2)(x - 5) = (x² + 4)(x² - 7x + 10) = x⁴ - 7x³ + 14x² - 28x + 40
This confirms that our factorization is indeed correct. We have successfully factored the polynomial f(x) completely.
Conclusion
Factoring polynomial functions completely, especially those with complex roots, requires a systematic approach and a solid understanding of key theorems like the Conjugate Roots Theorem. By applying the theorem, performing polynomial division, and factoring quadratic expressions, we can break down complex polynomials into manageable linear factors. In this article, we successfully factored the polynomial function f(x) = x⁴ - 7x³ + 14x² - 28x + 40 completely, given that -2i is a zero. The complete factorization is:
f(x) = (x + 2i)(x - 2i)(x - 2)(x - 5)
This step-by-step guide demonstrates the power of these techniques in unraveling the structure of polynomial functions. Mastering these methods will empower you to tackle a wide range of polynomial factorization problems.
By understanding the interplay between zeros, factors, and the Conjugate Roots Theorem, you can confidently navigate the world of polynomial factorization and gain a deeper appreciation for the elegance and interconnectedness of mathematics. Remember to always double-check your work and practice regularly to hone your skills. Factoring polynomials can be challenging, but with the right tools and a methodical approach, you can achieve success.