Factoring Polynomials Into Linear And Irreducible Quadratic Factors

In the realm of algebra, factoring polynomials stands as a fundamental technique, enabling us to decompose complex expressions into simpler, more manageable components. This process holds significant importance in various mathematical domains, including solving equations, simplifying expressions, and gaining deeper insights into the behavior of polynomial functions. Our focus here is on expressing a given polynomial as a product of linear and irreducible quadratic factors, a representation that provides valuable information about the polynomial's roots and its overall structure. We will delve into the intricacies of this process, using the specific example of the polynomial f(x) = x⁴ + 3x³ - 17x² + 3x - 18 to illustrate the key steps and concepts involved. By mastering this technique, we unlock a powerful tool for unraveling the complexities of polynomial expressions and gaining a more profound understanding of their properties. The ability to factor polynomials into linear and irreducible quadratic factors is a cornerstone of algebraic manipulation, with far-reaching applications in mathematics and related fields. This method empowers us to dissect complex expressions into their fundamental building blocks, revealing the underlying structure and behavior of polynomial functions. In this comprehensive exploration, we will embark on a step-by-step journey to factor the given polynomial, unveiling the elegant interplay of algebraic techniques that leads to the desired factorization.

Understanding the Problem

The task at hand is to express the given polynomial, f(x) = x⁴ + 3x³ - 17x² + 3x - 18, as a product of linear and irreducible quadratic factors, all with real coefficients. This means we aim to rewrite the polynomial in the form:

f(x) = (ax + b)(cx + d)(ex² + fx + g)

where a, b, c, d, e, f, and g are real numbers, and the quadratic factor (ex² + fx + g) cannot be factored further into linear factors with real coefficients. This irreducibility condition implies that the discriminant of the quadratic factor, given by f² - 4eg, is negative. Before we dive into the solution, let's break down the core concepts and strategies involved in polynomial factorization. At the heart of polynomial factorization lies the quest to decompose a complex expression into simpler, more manageable components. This process not only simplifies the polynomial but also unveils crucial information about its roots and behavior. In our specific problem, we seek to express the given polynomial as a product of linear and irreducible quadratic factors, each playing a distinct role in shaping the overall polynomial. Linear factors, of the form (ax + b), directly correspond to real roots of the polynomial, providing immediate insights into where the polynomial intersects the x-axis. Irreducible quadratic factors, on the other hand, represent quadratic expressions that cannot be further factored into linear factors with real coefficients. These factors contribute to the polynomial's behavior without introducing additional real roots. The challenge lies in systematically identifying these linear and irreducible quadratic factors and piecing them together to reconstruct the original polynomial. The journey begins with exploring potential rational roots, employing techniques like the Rational Root Theorem to narrow down the possibilities. Once we identify a root, we can utilize synthetic division or polynomial long division to extract the corresponding linear factor, simplifying the polynomial's degree. This process may need to be repeated several times until we arrive at a quadratic expression that can be either factored into linear factors or determined to be irreducible. The irreducibility of a quadratic factor is readily determined by examining its discriminant, a value that reveals the nature of the quadratic's roots. A negative discriminant signals that the quadratic has no real roots, confirming its irreducibility over real numbers. By skillfully combining these techniques, we can systematically navigate the factorization process, ultimately expressing the polynomial as a product of linear and irreducible quadratic factors, each contributing to the polynomial's unique characteristics.

Finding Rational Roots

To begin, we can use the Rational Root Theorem to find potential rational roots of the polynomial. The Rational Root Theorem states that if a polynomial has a rational root p/q (where p and q are coprime integers), then p must be a factor of the constant term and q must be a factor of the leading coefficient. In our case, the constant term is -18 and the leading coefficient is 1. Therefore, the possible rational roots are the factors of -18, which are ±1, ±2, ±3, ±6, ±9, and ±18. We can test these potential roots by substituting them into the polynomial f(x). If f(r) = 0 for some rational number r, then r is a root of the polynomial, and (x - r) is a factor. Let's start by testing x = 1: f(1) = 1⁴ + 3(1)³ - 17(1)² + 3(1) - 18 = 1 + 3 - 17 + 3 - 18 = -28 ≠ 0 So, x = 1 is not a root. Next, let's try x = -1: f(-1) = (-1)⁴ + 3(-1)³ - 17(-1)² + 3(-1) - 18 = 1 - 3 - 17 - 3 - 18 = -40 ≠ 0 Thus, x = -1 is not a root either. Now, let's test x = 3: f(3) = (3)⁴ + 3(3)³ - 17(3)² + 3(3) - 18 = 81 + 81 - 153 + 9 - 18 = 0 This shows that x = 3 is a root, and therefore, (x - 3) is a factor of the polynomial. The journey of factoring polynomials often begins with the quest for rational roots, those elegant numbers that, when plugged into the polynomial, yield a result of zero. These roots serve as stepping stones, guiding us towards the linear factors that constitute the polynomial's building blocks. The Rational Root Theorem, a cornerstone of polynomial factorization, provides a systematic approach to identify potential rational roots. This theorem elegantly connects the polynomial's coefficients to its possible rational roots, narrowing down the search and transforming what could be an overwhelming task into a manageable endeavor. In essence, the Rational Root Theorem states that any rational root of the polynomial must be a fraction whose numerator is a factor of the constant term and whose denominator is a factor of the leading coefficient. By meticulously listing the factors of these coefficients, we create a comprehensive inventory of potential rational roots, each a candidate for unraveling the polynomial's structure. The next step involves testing these candidates, a process akin to detective work, where we substitute each potential root into the polynomial and observe the outcome. A zero result signals a successful identification, confirming the candidate as a root and revealing a corresponding linear factor. This linear factor, of the form (x - r), where r is the root, can then be extracted from the polynomial, simplifying the expression and paving the way for further factorization. The search for rational roots may require testing several candidates, but each successful find brings us closer to the ultimate goal of expressing the polynomial as a product of linear and irreducible quadratic factors. This systematic approach, guided by the Rational Root Theorem, transforms the daunting task of polynomial factorization into a methodical and rewarding exploration of algebraic structures.

Polynomial Division

Since we found that (x - 3) is a factor, we can perform polynomial division to divide f(x) by (x - 3). This will give us a quotient polynomial of degree 3. Performing polynomial long division or synthetic division, we get:

 x³ + 6x² + x + 6
x - 3 | x⁴ + 3x³ - 17x² + 3x - 18
 - (x⁴ - 3x³)
 6x³ - 17x²
 - (6x³ - 18x²)
 x² + 3x
 - (x² - 3x)
 6x - 18
 - (6x - 18)
 0

So, f(x) = (x - 3)(x³ + 6x² + x + 6). Now we need to factor the cubic polynomial x³ + 6x² + x + 6. Once we've identified a linear factor, the next step is to divide the original polynomial by this factor. This process, known as polynomial division, serves as a powerful tool for reducing the degree of the polynomial, making it more manageable for further factorization. By dividing out the known linear factor, we effectively peel away one layer of complexity, revealing a polynomial of lower degree that retains the remaining factors. Polynomial division can be performed using a couple of different methods, each with its own set of advantages. Long division, a familiar technique from arithmetic, provides a step-by-step approach to dividing polynomials, meticulously aligning terms and tracking remainders. This method is particularly useful when dealing with polynomials of higher degrees or when the divisor is not a simple linear factor. Alternatively, synthetic division offers a more streamlined approach for dividing by linear factors of the form (x - r). This method utilizes a compact notation and a series of arithmetic operations to efficiently compute the quotient and remainder. Regardless of the method chosen, the result of polynomial division is a quotient polynomial and a remainder. If the division is exact, meaning the remainder is zero, then the divisor is indeed a factor of the original polynomial, and the quotient represents the remaining polynomial expression to be factored. On the other hand, a non-zero remainder indicates that the divisor is not a factor, and we must explore other potential factors. The quotient polynomial obtained through polynomial division holds valuable information about the remaining factors of the original polynomial. Its degree is one less than the original polynomial, reflecting the extraction of the linear factor. By factoring the quotient polynomial, we can uncover additional linear and irreducible quadratic factors, ultimately leading to the complete factorization of the original polynomial.

Factoring the Cubic Polynomial

We can use factoring by grouping for x³ + 6x² + x + 6:

x³ + 6x² + x + 6 = x²(x + 6) + 1(x + 6) = (x² + 1)(x + 6)

Thus, f(x) = (x - 3)(x + 6)(x² + 1). The quadratic factor (x² + 1) is irreducible over real numbers because its discriminant (0² - 4(1)(1) = -4) is negative. Once we've reduced the polynomial's degree through division, the quest for factors often continues with the exploration of various factoring techniques tailored to the resulting polynomial. For cubic polynomials, such as the one we encountered after dividing by (x - 3), a technique known as factoring by grouping can prove particularly effective. Factoring by grouping involves strategically pairing terms within the polynomial, seeking common factors within each pair. By factoring out these common factors, we aim to reveal a shared binomial factor that can then be extracted from the entire expression. This process transforms the cubic polynomial into a product of simpler expressions, often including a quadratic factor that may be irreducible or further factorable. In our specific case, the cubic polynomial x³ + 6x² + x + 6 lends itself beautifully to factoring by grouping. By pairing the first two terms and the last two terms, we observe the potential for common factors to emerge. From the first pair, x³ + 6x², we can factor out x², leaving us with x²(x + 6). Similarly, from the second pair, x + 6, we can factor out 1, resulting in 1(x + 6). Now, a remarkable pattern emerges: both pairs share the binomial factor (x + 6). This shared factor acts as a bridge, connecting the two groups and enabling us to factor it out from the entire expression. By extracting (x + 6), we obtain the factored form (x² + 1)(x + 6), a significant milestone in our factorization journey. The cubic polynomial has now been expressed as the product of a linear factor, (x + 6), and a quadratic factor, (x² + 1). The linear factor directly corresponds to a real root of the polynomial, while the quadratic factor's nature remains to be investigated. The quadratic factor, (x² + 1), holds the key to understanding the polynomial's remaining roots. To determine whether it can be factored further into linear factors with real coefficients, we turn to the discriminant, a value that reveals the nature of a quadratic's roots. A negative discriminant signals that the quadratic has no real roots, confirming its irreducibility over real numbers. This irreducibility signifies that the quadratic factor contributes to the polynomial's behavior without introducing additional real roots, adding a layer of sophistication to the overall factorization. With the cubic polynomial successfully factored, we are one step closer to expressing the original polynomial as a product of linear and irreducible quadratic factors, a representation that unveils its intricate structure and behavior.

Final Answer

Therefore, the function can be written as a product of linear and irreducible quadratic factors as:

f(x) = (x - 3)(x + 6)(x² + 1)

So, the correct answer is B. (x - 3)(x + 6)(x² + 1). As we approach the final answer in our factorization journey, we gather the fruits of our efforts, piecing together the linear and irreducible quadratic factors that we have meticulously identified. The culmination of this process is the expression of the original polynomial as a product of these fundamental building blocks, a representation that unveils its intricate structure and behavior. Each factor plays a distinct role in shaping the polynomial's characteristics. Linear factors, of the form (ax + b), correspond directly to the polynomial's real roots, indicating where the polynomial intersects the x-axis. Irreducible quadratic factors, on the other hand, contribute to the polynomial's behavior without introducing additional real roots, adding a layer of complexity to its overall shape. The final factored form provides a comprehensive picture of the polynomial's roots and its overall algebraic structure. It allows us to easily identify the real roots, determine the polynomial's behavior near these roots, and understand how the irreducible quadratic factors influence its shape. In our specific example, we have successfully factored the polynomial f(x) = x⁴ + 3x³ - 17x² + 3x - 18 into the product (x - 3)(x + 6)(x² + 1). This factorization reveals the polynomial's real roots at x = 3 and x = -6, corresponding to the linear factors (x - 3) and (x + 6), respectively. The irreducible quadratic factor (x² + 1) indicates the presence of complex roots, which do not appear on the real number line. By carefully analyzing the factored form, we gain a profound understanding of the polynomial's behavior, its roots, and its overall algebraic structure. This knowledge empowers us to solve equations, sketch graphs, and delve deeper into the mathematical properties of the polynomial. The journey of polynomial factorization, culminating in the final answer, is a testament to the power of algebraic techniques in unraveling the complexities of mathematical expressions. It showcases how systematic approaches, guided by fundamental theorems and concepts, can transform seemingly intricate problems into manageable and insightful explorations.