Factoring Polynomials A Step By Step Guide To 18x³ - 120x² - 42x

In the realm of algebra, polynomial factorization stands as a fundamental technique, enabling us to break down complex expressions into simpler, more manageable components. This process not only simplifies algebraic manipulations but also provides valuable insights into the roots and behavior of polynomial functions. In this comprehensive guide, we will embark on a step-by-step journey to unravel the completely factored form of the polynomial 18x³ - 120x² - 42x. By mastering this technique, you'll gain a deeper understanding of polynomial structures and enhance your problem-solving prowess in algebra.

1. Identifying the Greatest Common Factor (GCF): The Foundation of Factoring

The cornerstone of polynomial factorization lies in identifying the greatest common factor (GCF), the largest factor that divides each term of the polynomial without leaving a remainder. Extracting the GCF effectively reduces the complexity of the polynomial, paving the way for further factorization. To determine the GCF of 18x³ - 120x² - 42x, we systematically analyze the coefficients and variable terms.

First, we consider the coefficients: 18, -120, and -42. The GCF of these numbers is the largest number that divides all three. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. Comparing these lists, we find that the greatest common factor of the coefficients is 6. In other words, 6 is the largest number that can evenly divide 18, 120, and 42. This forms a crucial part of our GCF for the entire polynomial.

Next, we turn our attention to the variable terms: x³, x², and x. Here, the GCF is the lowest power of x present in all terms. We have x raised to the power of 3 (x³), x raised to the power of 2 (x²), and x raised to the power of 1 (x). The lowest power of x among these is x¹ or simply x. Therefore, the GCF of the variable terms is x. By identifying this, we ensure that our factored form includes the common variable component that accurately represents the original polynomial.

Combining the GCF of the coefficients (6) and the GCF of the variable terms (x), we arrive at the overall GCF of the polynomial 18x³ - 120x² - 42x, which is 6x. This means that 6x is the largest expression that can divide each term of the polynomial evenly. Now that we have identified the GCF, we can proceed with factoring it out of the polynomial, which simplifies the expression and allows us to address the remaining quadratic expression more effectively. This step is essential in reducing the complexity of the polynomial and setting the stage for subsequent factoring techniques.

2. Factoring out the GCF: Simplifying the Polynomial

With the GCF, 6x, firmly in hand, we embark on the crucial step of factoring it out from the polynomial 18x³ - 120x² - 42x. This process involves dividing each term of the polynomial by the GCF and expressing the result as a product of the GCF and the remaining expression. This not only simplifies the polynomial but also unveils its underlying structure, making further factorization more accessible.

To factor out 6x from 18x³, we perform the division 18x³ ÷ 6x. Dividing the coefficients, 18 ÷ 6 equals 3. Dividing the variable terms, x³ ÷ x equals x². Thus, the result of dividing the first term by the GCF is 3x². This indicates that 3x² is the component of the new polynomial expression derived from the first term of the original polynomial.

Next, we factor out 6x from -120x². Dividing the coefficients, -120 ÷ 6 equals -20. Dividing the variable terms, x² ÷ x equals x. Consequently, the result of dividing the second term by the GCF is -20x. The negative sign is critical here, as it correctly represents the operation within the original polynomial, ensuring that the factored form accurately reflects the initial expression.

Finally, we factor out 6x from -42x. Dividing the coefficients, -42 ÷ 6 equals -7. Dividing the variable terms, x ÷ x equals 1. Hence, the result of dividing the third term by the GCF is -7. The absence of a variable in this term simplifies this component and completes the division process for all terms of the original polynomial.

Combining these results, we can rewrite the polynomial 18x³ - 120x² - 42x as the product of the GCF and the expression obtained after division: 6x(3x² - 20x - 7). This factored form not only simplifies the original polynomial but also transforms it into a more manageable expression for further analysis. By extracting the GCF, we have effectively reduced the degree of the polynomial inside the parentheses, making it easier to handle. The expression 3x² - 20x - 7 is a quadratic expression, which we will address in the next step to fully factor the original polynomial. This step highlights the significance of identifying and factoring out the GCF, which is essential for simplifying complex polynomials and preparing them for subsequent factorization techniques.

3. Factoring the Quadratic Expression: Unveiling the Roots

After extracting the GCF, we arrive at the quadratic expression 3x² - 20x - 7, a crucial piece in our quest for the completely factored form of the original polynomial. Factoring a quadratic expression involves decomposing it into two binomial factors, a process that reveals the roots or zeros of the quadratic function. To tackle this, we employ a systematic approach that ensures accuracy and efficiency.

The goal is to express 3x² - 20x - 7 as a product of two binomials: (ax + b)(cx + d). This involves finding the appropriate values for a, b, c, and d that satisfy certain conditions derived from the coefficients of the quadratic expression. The product of the first terms of the binomials (ax and cx) must equal the first term of the quadratic (3x²), and the product of the last terms (b and d) must equal the last term of the quadratic (-7). Additionally, the sum of the outer and inner products (adx and bcx) must equal the middle term of the quadratic (-20x).

To find these values, we can use various methods, such as factoring by grouping or the quadratic formula. Here, we will use the factoring by grouping method. We need to find two numbers that multiply to the product of the leading coefficient (3) and the constant term (-7), which is -21, and add up to the middle coefficient (-20). These two numbers are -21 and 1 because (-21) * 1 = -21 and (-21) + 1 = -20. These numbers are essential for rewriting the middle term of the quadratic expression and facilitating factoring by grouping.

Using these numbers, we rewrite the middle term of the quadratic expression: 3x² - 20x - 7 becomes 3x² - 21x + x - 7. By breaking down the middle term (-20x) into (-21x + x), we create a four-term polynomial that is conducive to factoring by grouping. This manipulation sets the stage for the subsequent grouping and factoring steps.

Next, we group the first two terms and the last two terms: (3x² - 21x) + (x - 7). This grouping allows us to identify common factors within each pair of terms, which is a critical step in the factoring process. Factoring out the GCF from each group, we factor 3x from the first group and 1 from the second group: 3x(x - 7) + 1(x - 7). The common binomial factor (x - 7) indicates that we are on the right track. This shared factor is the key to simplifying the expression further.

Now, we factor out the common binomial factor (x - 7) from the entire expression: (3x + 1)(x - 7). This step effectively combines the terms and represents the quadratic expression as a product of two binomials. Thus, we have successfully factored the quadratic expression 3x² - 20x - 7 into (3x + 1)(x - 7). This decomposition is a crucial step in obtaining the completely factored form of the original polynomial, revealing the linear factors that constitute the quadratic expression. By identifying these factors, we gain deeper insights into the structure and behavior of the quadratic function represented by the expression.

4. The Completely Factored Form: Putting it All Together

Having meticulously extracted the GCF and factored the resulting quadratic expression, we now stand at the final step: assembling the completely factored form of the polynomial 18x³ - 120x² - 42x. This comprehensive factorization represents the polynomial as a product of its simplest factors, providing a profound understanding of its structure and behavior.

Recall that we initially identified the GCF as 6x and subsequently factored it out of the polynomial, yielding 6x(3x² - 20x - 7). This initial step was crucial in simplifying the polynomial and making it more amenable to further factorization. It reduced the complexity by removing the common factor present in each term, which set the stage for addressing the remaining quadratic expression.

Next, we successfully factored the quadratic expression 3x² - 20x - 7 into (3x + 1)(x - 7). This factorization involved breaking down the quadratic expression into its binomial factors, which are essential for understanding the roots and behavior of the polynomial. By expressing the quadratic expression in this form, we revealed the linear components that constitute the original polynomial.

To obtain the completely factored form, we combine the GCF with the factored quadratic expression. This involves multiplying the GCF, 6x, with the two binomial factors (3x + 1) and (x - 7). By bringing these components together, we achieve the ultimate representation of the polynomial in its most simplified form.

Therefore, the completely factored form of the polynomial 18x³ - 120x² - 42x is 6x(3x + 1)(x - 7). This expression represents the polynomial as a product of three factors: the monomial 6x and the binomials (3x + 1) and (x - 7). This factorization not only simplifies the polynomial but also provides valuable insights into its roots and behavior. The factors directly correspond to the roots of the polynomial, which are the values of x that make the polynomial equal to zero. Understanding these roots is critical for solving equations and analyzing the graph of the polynomial function.

The factored form also reveals the structure of the polynomial, showing how it is composed of simpler components. This can be particularly useful in various algebraic manipulations and applications, such as simplifying rational expressions, solving polynomial equations, and graphing polynomial functions. The completely factored form is, therefore, a powerful tool in algebra, providing both a simplified representation and a deeper understanding of the polynomial.

Conclusion: Mastering Polynomial Factorization

Through this detailed exploration, we have successfully unveiled the completely factored form of the polynomial 18x³ - 120x² - 42x, which is 6x(3x + 1)(x - 7). This journey has underscored the importance of systematic techniques in polynomial factorization, including identifying the GCF and factoring quadratic expressions.

By mastering these techniques, you equip yourself with valuable tools for simplifying algebraic expressions, solving polynomial equations, and gaining a deeper understanding of polynomial functions. Polynomial factorization is a fundamental concept in algebra, with applications extending to calculus and beyond. It enables us to analyze and manipulate complex expressions, making it an indispensable skill for anyone pursuing mathematical studies or careers.

The ability to factor polynomials not only simplifies mathematical problems but also enhances problem-solving skills in various contexts. Whether you are a student mastering algebraic concepts or a professional applying mathematical principles, the techniques discussed in this guide will serve as a solid foundation for your endeavors. Continue practicing and applying these methods, and you will find yourself increasingly confident and proficient in the art of polynomial factorization. Through consistent effort, you can effectively tackle a wide range of polynomial problems and gain a profound appreciation for the elegance and power of algebraic manipulation.