Prime Polynomial Product For 30x³ - 5x² - 60 A Detailed Solution

by ADMIN 65 views

In the realm of algebra, understanding the factorization of polynomials is crucial. Polynomial factorization involves expressing a polynomial as a product of simpler polynomials, often prime polynomials that cannot be factored further. In this comprehensive exploration, we will dissect the polynomial 30x³ - 5x² - 60 to determine its equivalent product of prime polynomials. This process requires a systematic approach, combining techniques like factoring out common factors and recognizing standard polynomial patterns. The correct identification of prime polynomial factors not only simplifies the expression but also provides deeper insights into the polynomial's structure and behavior. Through meticulous factorization, we aim to match the given polynomial with one of the provided options, thereby revealing the underlying prime polynomial composition.

The journey of factoring 30x³ - 5x² - 60 begins with identifying the greatest common factor (GCF) among all the terms. Factoring out the GCF is the first and often most crucial step in simplifying a polynomial. It sets the stage for further factorization by reducing the coefficients and potentially the degree of the polynomial. In this case, the numerical coefficients are 30, -5, and -60. By finding the GCF, we effectively reduce the complexity of the polynomial, making it easier to handle. This initial step not only simplifies the expression but also makes subsequent factoring steps more manageable. It's a foundational technique that every algebra student must master to tackle more complex polynomial expressions. Furthermore, the GCF helps in identifying potential roots or zeros of the polynomial, which are critical in various mathematical applications.

Moving forward, the reduced polynomial will be examined for recognizable patterns, such as the difference of squares, the sum or difference of cubes, or simple quadratic trinomials. Recognizing these patterns is essential for efficient factorization. For instance, if we encounter a difference of squares (a² - b²), we know it can be factored as (a + b)(a - b). Similarly, understanding the patterns for the sum and difference of cubes is vital for their factorization. The quadratic trinomials, in particular, require a keen eye to identify coefficients that allow for factoring into two binomials. This pattern recognition process is not just about memorizing formulas; it's about developing a mathematical intuition for how different algebraic expressions can be manipulated. By mastering these techniques, we equip ourselves with a powerful toolkit for simplifying and solving a wide range of polynomial equations. The ability to recognize and apply these patterns is a cornerstone of algebraic problem-solving.

After identifying any patterns, the next step involves expressing the polynomial as a product of its prime factors. Prime factors are polynomials that cannot be factored further. Factoring to prime polynomials ensures that we have simplified the expression to its most basic components. For quadratic expressions, this often means finding two binomials that, when multiplied, yield the original quadratic. This may involve trial and error or the use of the quadratic formula to find the roots, which then help in constructing the factors. The process of identifying prime factors is akin to finding the prime number decomposition of an integer – it's about breaking down the expression into its fundamental building blocks. By expressing the polynomial in terms of its prime factors, we gain a complete understanding of its structure, which is essential for solving equations, graphing functions, and various other applications in mathematics and science. This step solidifies the foundation for advanced algebraic manipulations.

Finally, after complete factorization, we compare our result with the given options to find the matching product of prime polynomials. Comparing the results with the provided choices is a critical step to ensure accuracy. Each option represents a potential factorization, and by comparing, we verify that our factored form is equivalent to one of the given options. This involves careful attention to the signs, coefficients, and powers of the variables in each term. It's not just about finding a factorization; it's about finding the correct factorization that matches the given expression. This process often requires expanding the factored form back to the original polynomial to confirm their equivalence. This comparative analysis reinforces the understanding of polynomial multiplication and factorization, ensuring that the identified factors are indeed the correct ones. This final step validates the entire factoring process and provides confidence in the solution.

Detailed Solution for Factoring 30x³ - 5x² - 60

Let’s delve into the step-by-step solution of factoring the polynomial 30x³ - 5x² - 60. This involves a methodical approach that begins with identifying the greatest common factor, proceeding to recognize patterns, and ultimately expressing the polynomial as a product of prime factors. This meticulous process not only provides the solution but also deepens the understanding of polynomial factorization.

  1. Identifying the Greatest Common Factor (GCF): The first step in factoring any polynomial is to find the greatest common factor (GCF) of all its terms. This simplifies the polynomial and makes subsequent factoring easier. In the polynomial 30x³ - 5x² - 60, we look for the largest number that divides all the coefficients evenly. The coefficients are 30, -5, and -60. The GCF of these numbers is 5. Additionally, we examine the variable terms. Here, we have x³ and x², but the constant term -60 has no x, so x is not a common factor. Thus, the GCF for the entire polynomial is 5. Factoring out the GCF involves dividing each term by 5 and writing the polynomial as 5 times the resulting expression. This step is crucial as it reduces the complexity of the polynomial, setting the stage for further factorization. The GCF acts as a bridge, connecting the original polynomial to a simpler, more manageable form.

  2. Factoring out the GCF: Now that we have identified the GCF as 5, we factor it out of the polynomial: 30x³ - 5x² - 60 = 5(6x³ - x² - 12). This step transforms the original polynomial into a product of the GCF and a new polynomial. The new polynomial 6x³ - x² - 12 is simpler to work with because the coefficients are smaller. Factoring out the GCF is a pivotal step as it simplifies the original polynomial into a more manageable form. This simplification is not just about aesthetics; it's about making the polynomial easier to factorize further. By reducing the magnitude of the coefficients, we reduce the number of potential factors, making the factoring process more efficient. This step also highlights the importance of recognizing and applying fundamental algebraic techniques.

  3. Analyzing the Remaining Polynomial: The next step is to analyze the polynomial inside the parentheses, which is 6x³ - x² - 12. This polynomial is a cubic polynomial, but it does not fit any common factoring patterns like the difference of squares or the sum/difference of cubes directly. Therefore, we need to look for other factorization techniques. Sometimes, we might consider factoring by grouping, synthetic division, or the rational root theorem. However, in this case, let's carefully consider the given options to guide our next steps. Analyzing the remaining polynomial is crucial as it sets the direction for further factorization. This step involves careful observation and consideration of various factoring techniques. It's a critical juncture where we decide which path to take based on the polynomial's structure and characteristics. The analysis may reveal hidden patterns or suggest a particular approach, such as factoring by grouping or using the rational root theorem. This decision-making process is a key aspect of algebraic problem-solving.

  4. Considering the Given Options: The given options are:

    • A. 5(2x² - 3)(3x + 4)
    • B. x(10x + 3)(3x - 4)
    • C. 5x(2x - 3)(3x + 4)
    • D. 5x(2x + 3)(3x - 4)

    We can see that option A has a factor of 5, which matches our GCF. The other options either have 'x' factored out, which isn't possible in our case, or do not have the correct GCF. Considering the given options is a strategic approach to polynomial factorization. It allows us to leverage the information provided in the multiple-choice answers to guide our factoring process. By examining the options, we can identify clues about the potential factors and narrow down the possibilities. This approach is particularly useful when dealing with complex polynomials where direct factorization might be challenging. The options serve as a roadmap, helping us navigate the factoring process more efficiently.

  5. Expanding Option A to Verify: Let's expand option A to verify if it matches our polynomial after factoring out the GCF: 5(2x² - 3)(3x + 4) = 5(6x³ + 8x² - 9x - 12)

    This expansion does not directly match our polynomial 6x³ - x² - 12 inside the parentheses. However, this discrepancy indicates a need to revisit our approach or identify potential errors. Expanding option A is a crucial verification step. It allows us to check if the proposed factorization is indeed equivalent to the original polynomial. If the expanded form does not match, it signals that the factorization is incorrect or incomplete. This step serves as a safety net, preventing us from settling on a wrong answer. It reinforces the importance of thoroughness and accuracy in algebraic manipulations.

  6. Revisiting the Polynomial and Options: We made an error in our GCF factoring. The constant term should have been -60 / 5 = -12, which is correct. However, expanding option A gives us:

    5(2x² - 3)(3x + 4) = 5(6x³ + 8x² - 9x - 12)

    Our polynomial after factoring out the 5 was 6x³ - x² - 12. We see a mismatch. Let’s re-examine the options and try to correct our approach. Revisiting the polynomial and options is a critical step when encountering discrepancies. It involves retracing our steps to identify potential errors or oversights. This process may lead us to reconsider our initial assumptions or explore alternative factoring techniques. It's a testament to the iterative nature of problem-solving, where persistence and attention to detail are paramount. This step reinforces the importance of critical thinking and analytical skills in mathematics.

  7. Correcting the Factoring Process: The error lies in assuming a direct match after factoring out the GCF. We need to factor the cubic further. Let's re-evaluate option A:

    5(2x² - 3)(3x + 4) = 5(6x³ + 8x² - 9x - 12)

    Our polynomial is 30x³ - 5x² - 60. If we divide this by 5, we get:

    6x³ - x² - 12

    We can see that expanding (2x² - 3)(3x + 4) does not give us 6x³ - x² - 12. Correcting the factoring process is essential to arrive at the correct solution. It involves identifying and rectifying any errors made during the factorization process. This may require revisiting previous steps, re-evaluating assumptions, or exploring alternative approaches. It's a process of refinement and precision, where attention to detail is crucial. This step underscores the importance of perseverance and a systematic approach to problem-solving.

  8. Trying a Different Approach: Synthetic Division/Rational Root Theorem: Let's consider a different approach since direct factoring isn't apparent. We can use the Rational Root Theorem to find possible rational roots of the cubic 6x³ - x² - 12. The possible rational roots are factors of -12 divided by factors of 6, which include ±1, ±2, ±3, ±4, ±6, ±12 divided by ±1, ±2, ±3, ±6. Trying x = -4/3:

    6(-4/3)³ - (-4/3)² - 12 = 6(-64/27) - 16/9 - 12 ≠ 0

    Trying x = 3/2:

    6(3/2)³ - (3/2)² - 12 = 6(27/8) - 9/4 - 12 = 81/4 - 9/4 - 48/4 = 24/4 = 6 ≠ 0

    Let's try a different factorization approach. Considering synthetic division and the rational root theorem is a valuable strategy when direct factoring methods are not readily apparent. This approach involves identifying potential rational roots of the polynomial and using synthetic division to test these roots. If a root is found, it can be used to factor the polynomial further. The rational root theorem provides a systematic way to identify potential roots, making the factoring process more efficient. This technique is particularly useful for higher-degree polynomials where factoring can be challenging.

  9. Expanding the Correct Option: After careful consideration and trying different methods, expanding option A:

    5(2x² - 3)(3x + 4) = 5(6x³ + 8x² - 9x - 12) does not match our polynomial.

    We realize that none of the options seem to directly match our factored form. There must be a misunderstanding in the question or the options provided. Expanding the correct option is a crucial step in verifying the solution. It involves multiplying the factors in the chosen option to see if the result matches the original polynomial. This step serves as a final check, ensuring that the factorization is accurate. If the expanded form matches the original polynomial, it confirms the correctness of the solution. This verification process is a cornerstone of algebraic problem-solving.

  10. Identifying the Correct Answer: Let's re-examine the given polynomial: 30x³ - 5x² - 60. We factored out 5 to get 5(6x³ - x² - 12). After further analysis, we find that none of the provided options match the correct factorization. It's possible there may be a typo in the options or the original polynomial. If we had the option 5(6x³ - x² - 12), that would be the correct answer. Identifying the correct answer involves a careful comparison of the factored form with the given options. It's a process of elimination and verification, where we look for the option that matches our factorization. If none of the options match, it may indicate an error in the problem statement or the options themselves. In such cases, it's important to acknowledge the discrepancy and communicate the issue.

Conclusion

In conclusion, the process of factoring the polynomial 30x³ - 5x² - 60 has led us through various algebraic techniques, including identifying the greatest common factor, considering standard polynomial patterns, and employing methods like synthetic division and the rational root theorem. Despite our efforts, we found that none of the provided options perfectly match the factored form. This discrepancy suggests a potential issue with the options or the original problem statement. While we couldn't definitively select one of the given answers, the detailed exploration has provided a comprehensive understanding of polynomial factorization and the importance of meticulous verification. This journey underscores the value of a systematic approach and the critical role of accuracy in mathematical problem-solving.