Factoring -4x³ + 44x² + 48x A Step-by-Step Guide
Factoring polynomials is a fundamental concept in algebra, enabling us to simplify complex expressions and solve equations. In this comprehensive guide, we will delve into the process of factoring the polynomial expression -4x³ + 44x² + 48x, providing a step-by-step approach to break it down into its simplest factors. This meticulous approach will not only aid in understanding the underlying principles of factoring but also equip you with the skills to tackle similar algebraic challenges with confidence. Let's embark on this journey of algebraic exploration, unraveling the intricacies of polynomial factorization.
Identifying the Greatest Common Factor (GCF)
When embarking on the journey of factoring polynomials, the initial and arguably most crucial step involves identifying the Greatest Common Factor (GCF). The GCF, as the name suggests, is the largest factor that is common to all terms within the polynomial. Recognizing and extracting the GCF simplifies the polynomial, making subsequent factoring steps significantly easier. In our specific case, the polynomial is -4x³ + 44x² + 48x. Let's meticulously examine each term to unveil the GCF.
First, we turn our attention to the coefficients: -4, 44, and 48. By carefully analyzing these numerical values, we can discern that the largest number that divides evenly into all three is 4. However, since the leading coefficient is negative, it is conventional to factor out a negative number. Therefore, we'll consider -4 as a potential component of the GCF. Next, we shift our focus to the variable component. We observe that each term contains 'x' raised to a power. The lowest power of 'x' present across all terms is 'x' to the power of 1 (or simply 'x'). This indicates that 'x' is indeed a common factor. Combining our observations, we conclude that the GCF for the polynomial -4x³ + 44x² + 48x is -4x. This initial step of identifying the GCF is paramount, as it lays the groundwork for the subsequent stages of factoring. By extracting the GCF, we effectively reduce the complexity of the polynomial, paving the way for a more manageable factoring process. This careful approach ensures that we tackle the factorization methodically, enhancing our understanding and accuracy.
Factoring out the GCF
Having successfully identified the Greatest Common Factor (GCF) as -4x, the next pivotal step in our factoring endeavor involves extracting this GCF from the polynomial -4x³ + 44x² + 48x. This process is akin to reverse distribution, where we divide each term of the polynomial by the GCF and place the GCF outside the parentheses. By meticulously executing this step, we effectively simplify the polynomial, making it more amenable to further factorization.
Let's methodically divide each term of the polynomial by -4x: -4x³/(-4x) = x². Here, we observe that dividing -4x³ by -4x yields x², a crucial component of the factored expression. Next, we divide the second term: 44x²/(-4x) = -11x. This division reveals that 44x² divided by -4x results in -11x, adding another piece to our factoring puzzle. Finally, we address the last term: 48x/(-4x) = -12. Dividing 48x by -4x gives us -12, completing the division process for all terms. Now, we assemble the results within the parentheses, placing the GCF, -4x, outside. This yields the expression -4x(x² - 11x - 12). Factoring out the GCF is a critical juncture in the factoring process. It not only simplifies the polynomial but also unveils the underlying structure, making subsequent factoring steps more transparent. By meticulously executing this step, we ensure that we are working with a reduced and more manageable expression, enhancing our chances of successfully factoring the polynomial.
Factoring the Quadratic Expression
Following the extraction of the Greatest Common Factor (GCF), our attention now shifts to the quadratic expression nestled within the parentheses: x² - 11x - 12. Factoring quadratic expressions is a cornerstone of algebra, and mastering this skill is crucial for simplifying polynomials and solving equations. To factor this quadratic, we embark on a quest to find two numbers that satisfy specific criteria. These numbers must not only multiply to give the constant term (-12) but also add up to the coefficient of the linear term (-11). This delicate balance is key to unlocking the factors of the quadratic expression.
Let's systematically explore pairs of factors of -12. We can start by considering the pairs: 1 and -12, -1 and 12, 2 and -6, -2 and 6, 3 and -4, and -3 and 4. Among these pairs, we seek the one that sums up to -11. A careful examination reveals that the pair 1 and -12 perfectly fits this criterion, as 1 multiplied by -12 yields -12, and 1 plus -12 equals -11. With these numbers in hand, we can now rewrite the quadratic expression in its factored form. The factored form is expressed as (x + 1)(x - 12). This transformation is a significant step forward, as it breaks down the quadratic expression into two linear factors. Factoring the quadratic expression is an art that requires practice and a keen eye for numerical relationships. By mastering this technique, we equip ourselves with a powerful tool for simplifying polynomials and solving equations. The ability to identify the correct pair of numbers that satisfy the multiplication and addition criteria is the key to unlocking the factors of the quadratic expression.
Writing the Final Factored Form
Having successfully factored the quadratic expression, we now stand at the threshold of completing our factoring journey. The penultimate step involves assembling all the pieces we've meticulously crafted thus far to construct the final factored form of the polynomial. We must not forget the Greatest Common Factor (GCF) that we initially extracted, as it plays an integral role in the complete factorization. By carefully piecing together the GCF and the factors of the quadratic expression, we arrive at the ultimate representation of the polynomial in its simplest factored form.
Recall that we extracted the GCF, -4x, in the initial stages of our factoring endeavor. Subsequently, we factored the quadratic expression x² - 11x - 12 into (x + 1)(x - 12). Now, we bring these two components together, combining the GCF with the factored quadratic. This synthesis yields the final factored form of the polynomial -4x³ + 44x² + 48x, which is expressed as -4x(x + 1)(x - 12). This final expression represents the polynomial in its most simplified and factored state. Writing the final factored form is the culmination of our efforts, a testament to our understanding of factoring techniques. It showcases our ability to dissect a complex polynomial and represent it as a product of its simplest factors. This skill is not only valuable in algebra but also serves as a foundation for more advanced mathematical concepts. By meticulously assembling the GCF and the factors of the quadratic, we ensure that our final factored form is both accurate and complete.
Verification and Conclusion
Before we declare our factoring endeavor complete, it is prudent to engage in a crucial step: verification. Verifying our factored form ensures that we have accurately represented the original polynomial. This process involves expanding the factored form and comparing the result with the original polynomial. If the expanded form matches the original polynomial, we can confidently assert that our factoring is correct. This final check not only validates our work but also reinforces our understanding of the relationship between factored and expanded forms.
To verify our factored form, -4x(x + 1)(x - 12), we embark on the expansion process. First, we expand the two binomial factors: (x + 1)(x - 12). This expansion yields x² - 12x + x - 12, which simplifies to x² - 11x - 12. Next, we distribute the -4x across the terms of the quadratic expression: -4x(x² - 11x - 12). This distribution results in -4x³ + 44x² + 48x. A careful comparison reveals that this expanded form perfectly matches our original polynomial, -4x³ + 44x² + 48x. This confirmation solidifies the accuracy of our factoring process. In conclusion, we have successfully factored the polynomial -4x³ + 44x² + 48x into its simplest form: -4x(x + 1)(x - 12). This journey through factoring has not only honed our algebraic skills but also deepened our appreciation for the elegance and precision of mathematical manipulation. Factoring polynomials is a fundamental skill that empowers us to simplify complex expressions and solve equations with confidence.
In summary, the journey of factoring the polynomial expression -4x³ + 44x² + 48x underscores the systematic approach required in algebraic manipulations. Starting with the crucial step of identifying and extracting the Greatest Common Factor (GCF), we simplified the expression, paving the way for subsequent factoring. The subsequent factoring of the quadratic expression involved a meticulous search for two numbers that satisfied specific multiplication and addition criteria, ultimately leading to its decomposition into linear factors. The culmination of these efforts was the assembly of the final factored form, a testament to our understanding of factoring techniques. Finally, the verification step served as a validation of our work, reinforcing the accuracy of our factoring process. This comprehensive exploration not only enhances our algebraic proficiency but also instills a deeper appreciation for the elegance and precision of mathematical problem-solving.
