Factor Trinomial $-x^2 + 2x + 48$ A Step-by-Step Guide
In the realm of mathematics, factorization stands as a fundamental technique for simplifying expressions and solving equations. Trinomials, which are quadratic expressions with three terms, are frequently encountered, and mastering their factorization is crucial for success in algebra and beyond. In this comprehensive guide, we will delve into the factorization of the trinomial $-x^2 + 2x + 48$, exploring the underlying principles and step-by-step procedures to arrive at the correct solution. We will analyze the structure of the trinomial, identify the key factors, and demonstrate how to express the trinomial as a product of binomials. By the end of this exploration, you will have a solid grasp of trinomial factorization and be well-equipped to tackle similar problems with confidence.
Understanding Trinomial Factorization
Before we embark on the factorization journey, let's establish a clear understanding of what trinomial factorization entails. A trinomial, in its general form, is expressed as $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Factorization, in essence, is the process of decomposing a trinomial into the product of two binomials, which are expressions with two terms. These binomials, when multiplied together, should yield the original trinomial. The goal of factorization is to rewrite the trinomial in a more simplified form, making it easier to solve equations, analyze graphs, and perform other mathematical operations.
The Significance of Factorization
Factorization is not merely an abstract mathematical exercise; it holds immense practical significance in various fields. In algebra, factorization is indispensable for solving quadratic equations, simplifying algebraic expressions, and determining the roots of polynomials. In calculus, factorization plays a vital role in finding limits, derivatives, and integrals. Moreover, factorization finds applications in computer science, engineering, and economics, where it is used to model and solve real-world problems. Mastering factorization empowers individuals to approach complex mathematical challenges with greater proficiency and confidence.
Deconstructing the Trinomial $-x^2 + 2x + 48$
Now, let's focus our attention on the trinomial $-x^2 + 2x + 48$ and dissect its structure to pave the way for factorization. The first term, $-x^2$, indicates a negative leading coefficient, which is a crucial observation that will guide our factorization strategy. The middle term, $2x$, represents the linear term, while the last term, $48$, is the constant term. Our objective is to find two binomials that, when multiplied, will produce this trinomial. To achieve this, we need to identify the factors of the constant term and strategically combine them to match the coefficient of the linear term.
Factoring Out the Negative Sign
The presence of the negative leading coefficient can initially pose a challenge. To simplify the factorization process, we can factor out a $-1$ from the entire trinomial. This transforms the expression into $-1(x^2 - 2x - 48)$. By factoring out the negative sign, we convert the trinomial inside the parentheses into a more manageable form with a positive leading coefficient. This step is crucial as it allows us to apply standard factorization techniques more effectively. The negative sign, however, should not be forgotten; it will play a significant role in the final factored form.
Unveiling the Factors of -48
The next step in our factorization journey involves identifying the factors of the constant term, which is now $-48$ after factoring out the negative sign. Factors are numbers that, when multiplied together, yield the given number. In this case, we need to find pairs of factors that multiply to $-48$. Since the constant term is negative, one factor in each pair must be positive, and the other must be negative. This is a critical observation that narrows down our search for the correct factors.
Systematic Factor Identification
To systematically identify the factors of $-48$, we can start by listing the factor pairs: $(1, -48)$, $(-1, 48)$, $(2, -24)$, $(-2, 24)$, $(3, -16)$, $(-3, 16)$, $(4, -12)$, $(-4, 12)$, $(6, -8)$, and $(-6, 8)$. Each of these pairs, when multiplied, results in $-48$. However, not all of these pairs will be suitable for our factorization purpose. We need to select the pair that, when combined, will produce the coefficient of the linear term, which is $-2$ in the trinomial $x^2 - 2x - 48$.
The Sum of Factors: The Key to the Linear Term
The connection between the factors of the constant term and the coefficient of the linear term is the cornerstone of trinomial factorization. The sum of the two factors must equal the coefficient of the linear term. In our case, we need to find a factor pair whose sum is $-2$. Examining the list of factor pairs we generated earlier, we can identify the pair $(6, -8)$ as the one that satisfies this condition. The sum of $6$ and $-8$ is indeed $-2$, which matches the coefficient of the linear term in the trinomial $x^2 - 2x - 48$. This discovery is a significant breakthrough, as it provides us with the crucial pieces needed to construct the binomial factors.
Constructing the Binomial Factors
With the factors $6$ and $-8$ identified, we can now construct the binomial factors of the trinomial $x^2 - 2x - 48$. The binomial factors will take the form $(x + p)(x + q)$, where $p$ and $q$ are the factors we found. Substituting $6$ and $-8$ for $p$ and $q$, respectively, we obtain the binomial factors $(x + 6)$ and $(x - 8)$. These binomials, when multiplied together, should yield the trinomial $x^2 - 2x - 48$. To verify this, we can perform the multiplication:
The multiplication confirms that our binomial factors are indeed correct.
Reintroducing the Negative Sign
We have successfully factored the trinomial $x^2 - 2x - 48$ into $(x + 6)(x - 8)$. However, we must not forget the negative sign that we factored out earlier. To obtain the complete factorization of the original trinomial, $-x^2 + 2x + 48$, we need to reintroduce the negative sign. This gives us the final factored form:
This is the ultimate factorization of the trinomial $-x^2 + 2x + 48$, expressing it as the product of a constant and two binomials.
The Correct Answer: Option D
Having successfully factored the trinomial, we can now compare our result with the options provided. Option D, $-1(x - 8)(x + 6)$, matches our factored form exactly. It is important to note that the order of the binomial factors does not affect the final result, as multiplication is commutative. Therefore, $-1(x + 6)(x - 8)$ is equivalent to $-1(x - 8)(x + 6)$. This confirms that Option D is the correct answer.
Distinguishing Correct Factorization from Incorrect Options
To further solidify our understanding, let's examine why the other options are incorrect. Option A, $-1(x + 8)(x + 6)$, is incorrect because the product of the factors $8$ and $6$ does not yield the constant term $-48$. Option B, $(x + 6)(x - 8)$, is incorrect because it does not include the negative sign that we factored out initially. Option C, $(-x + 6)(x + 8)$, is also incorrect because, when multiplied, it does not produce the original trinomial $-x^2 + 2x + 48$. By analyzing these incorrect options, we reinforce our understanding of the crucial steps involved in trinomial factorization and the importance of careful attention to detail.
Alternative Approaches to Factorization
While the method we have employed is a standard and effective technique for trinomial factorization, it is worth noting that alternative approaches exist. One such approach is the "AC method," which involves multiplying the leading coefficient ($a$) by the constant term ($c$) and then finding factors of the product that add up to the coefficient of the linear term ($b$). Another approach is the "trial and error" method, where we systematically try different combinations of binomial factors until we arrive at the correct factorization. However, the method we have presented, which focuses on identifying the factors of the constant term and their sum, is generally considered to be the most intuitive and efficient method for factoring trinomials.
Mastering the Art of Factorization
Factorization is a skill that improves with practice. The more trinomials you factor, the more proficient you will become in recognizing patterns, identifying factors, and constructing binomial factors. It is essential to approach factorization systematically, breaking down the problem into smaller, manageable steps. By understanding the underlying principles and practicing consistently, you can master the art of factorization and unlock a powerful tool for solving mathematical problems.
Conclusion: The Power of Factorization
In conclusion, the factorization of the trinomial $-x^2 + 2x + 48$ is $-1(x - 8)(x + 6)$, as demonstrated through our step-by-step analysis. We began by understanding the concept of trinomial factorization, emphasizing its significance in mathematics and various fields. We then dissected the trinomial, factored out the negative sign, identified the factors of the constant term, and strategically combined them to construct the binomial factors. By reintroducing the negative sign, we arrived at the complete factored form. This exploration highlights the power of factorization in simplifying expressions, solving equations, and gaining deeper insights into mathematical relationships. With a solid grasp of factorization techniques, you are well-equipped to tackle a wide range of mathematical challenges with confidence and precision.
What is the factorization process to solve the trinomial $-x^2+2 x+48$?
Factor Trinomial $-x^2 + 2x + 48$: A Step-by-Step Guide
