Factoring -7x² - 4x + 20 A Step By Step Solution
Factoring trinomials can often seem like a daunting task, but with a systematic approach and a clear understanding of the underlying principles, it becomes a manageable and even rewarding exercise. In this article, we will embark on a journey to factor the trinomial -7x² - 4x + 20, breaking down each step and illuminating the path to the correct factorization. Our focus will be on empowering you with the knowledge and skills to confidently tackle similar problems in the future. We will analyze the given options and provide a detailed explanation of why one is correct, while clarifying the errors in the others. This method ensures you understand the process, not just the answer.
The challenge of factoring -7x² - 4x + 20 lies in the negative leading coefficient and the specific combination of coefficients. These elements necessitate careful consideration of signs and factors. Through a methodical examination, we will unravel this challenge and reveal the solution. First, let’s try and understand how to factor a quadratic expression, in general, and then we will look at the specifics of this problem. This general understanding is crucial as it equips you with the ability to solve a wide variety of factoring problems.
Before diving into the specifics of this problem, it is important to understand the general principles of factoring quadratic expressions. A quadratic expression is generally represented as ax² + bx + c, where a, b, and c are constants. Factoring involves expressing this quadratic expression as a product of two binomials. The key is to find two numbers that multiply to give ac and add up to b. This might sound complex, but we will break it down step by step. Factoring is essential in many areas of mathematics, including solving equations, simplifying expressions, and analyzing graphs. Mastering this skill will significantly enhance your mathematical capabilities.
Understanding the concept of factoring is not just about getting the correct answer; it's about developing a deep understanding of the structure of algebraic expressions. By learning to factor, you will start to see patterns and relationships between different parts of an expression. This will make it easier to manipulate algebraic expressions and solve equations. For instance, in calculus, factoring is frequently used to simplify expressions before integration or differentiation. In higher-level mathematics, factoring concepts extend to polynomials of higher degrees and even abstract algebraic structures. Therefore, mastering factoring is a fundamental building block for advanced mathematical studies.
To begin, we must acknowledge the negative coefficient of the x² term, which is -7. This negative sign significantly impacts the factoring process. The most effective strategy is to factor out a -1 from the entire trinomial. This simplifies the expression and allows us to work with a positive leading coefficient, which makes the subsequent steps easier to manage. Factoring out -1 gives us: -1(7x² + 4x - 20). This simple step is crucial because it sets the stage for factoring the quadratic inside the parentheses more effectively. Without this step, you might find yourself struggling with the signs and coefficients.
Factoring out the -1 is a strategic move that streamlines the process. It allows us to focus on factoring the expression 7x² + 4x - 20, which has a positive leading coefficient. This eliminates the potential for sign errors that can easily occur when dealing with negative leading coefficients. Furthermore, by factoring out the -1, we are essentially simplifying the problem into a more manageable form. This technique is widely used in algebra and is particularly useful when dealing with complex expressions. It demonstrates the importance of strategic simplification in mathematical problem-solving.
After factoring out the -1, we are left with the trinomial 7x² + 4x - 20. This trinomial now has a positive leading coefficient, which simplifies the factoring process. To factor this, we need to find two numbers that multiply to the product of the leading coefficient (7) and the constant term (-20), which is -140, and add up to the middle coefficient (4). This is the core of the factoring process for trinomials of this form. Finding these two numbers might require some trial and error, but it is a crucial step in unlocking the factors of the trinomial. The ability to identify these numbers is what distinguishes successful factoring from mere guessing.
The challenge now is to find these two numbers. This often involves listing the factors of -140 and checking which pair adds up to 4. The factors of 140 are: 1 and 140, 2 and 70, 4 and 35, 5 and 28, 7 and 20, 10 and 14. Since we need a negative product (-140), one of the numbers must be negative. By trying different combinations, we find that 14 and -10 satisfy our conditions: 14 * -10 = -140 and 14 + (-10) = 4. These are the magic numbers that will help us break down the middle term and factor the trinomial. Identifying these numbers is a critical step, and it requires a good understanding of number properties and factorization.
The process of finding the correct factors is not always straightforward, especially when dealing with larger numbers. However, a systematic approach can make the task easier. One strategy is to start with the smaller factors and work your way up. Another technique is to focus on the prime factorization of the number, which can help you identify all possible factor pairs. In our case, the prime factorization of 140 is 2² * 5 * 7, which allows us to systematically explore different combinations. Patience and persistence are key in this process. With practice, you will develop an intuition for identifying the correct factors more quickly.
With these numbers identified, we rewrite the middle term (4x) as the sum of 14x and -10x. This transforms the trinomial into a four-term expression: 7x² + 14x - 10x - 20. This rewriting is a crucial step in the factoring by grouping method. By splitting the middle term, we create a structure that allows us to factor out common factors from pairs of terms. This method is a powerful tool for factoring trinomials, especially those with leading coefficients other than 1. The key is to rewrite the middle term in a way that facilitates the identification of common factors in the subsequent grouping step.
Factoring by grouping is a versatile technique that can be applied to a wide range of polynomial expressions. The basic idea is to group terms together in pairs and then factor out the greatest common factor (GCF) from each pair. If the resulting expressions in the parentheses are the same, you can then factor out the common binomial factor. This method is particularly useful when dealing with expressions that have four or more terms. The success of factoring by grouping depends on correctly identifying the terms to group and the GCF to factor out. Practice with different types of expressions will help you master this technique.
Now, we factor by grouping. From the first two terms (7x² + 14x), we can factor out 7x, which gives us 7x(x + 2). From the last two terms (-10x - 20), we can factor out -10, which gives us -10(x + 2). Notice that both expressions now have a common factor of (x + 2). This is a critical point in the process, as it indicates that we have correctly chosen our factors. The next step is to factor out the common binomial factor (x + 2), which will lead us to the final factored form of the trinomial. Recognizing this common factor is key to successfully completing the factoring process.
Factoring out the common binomial (x + 2) gives us (7x - 10)(x + 2). Now, we must not forget the -1 we factored out at the beginning. So, the complete factorization is -1(7x - 10)(x + 2). This is one of the options provided, and it is the correct factorization of the trinomial. It’s important to remember the initial step of factoring out the -1, as this is often a source of error for students. Completing the factorization requires careful attention to detail and a systematic approach. By following the steps outlined above, you can confidently factor trinomials of this type.
Checking your factorization is an essential step in the problem-solving process. You can do this by multiplying the factors back together to see if you get the original trinomial. In our case, multiplying -1(7x - 10)(x + 2) should give us -7x² - 4x + 20. This verification step not only confirms the correctness of your solution but also reinforces your understanding of the factoring process. It’s a good practice to always check your work, especially in high-stakes situations like exams. This habit of verification will build your confidence and prevent careless errors.
Now, let's compare our result with the provided answer choices:
A. 7(x + 10)(-x + 2) B. -7(x - 5)(x + 2) C. -1(7x - 10)(x + 2) D. (-7x + 10)(x - 2)
We can clearly see that option C, -1(7x - 10)(x + 2), matches our factored form. Therefore, this is the correct answer. The other options are incorrect due to various reasons. Option A has incorrect factors, and option B has the wrong constant terms within the factors. Option D, when multiplied out, will result in a different trinomial than the one we started with. Analyzing the answer choices is a crucial step in the problem-solving process. It allows you to compare your result with the available options and identify the correct answer with confidence. This step also provides an opportunity to catch any errors you may have made during the factoring process.
Option A, 7(x + 10)(-x + 2), is incorrect because when expanded, it does not yield the original trinomial -7x² - 4x + 20. The factors (x + 10) and (-x + 2) would result in a different quadratic expression. This highlights the importance of meticulously checking each factor and the overall result. An incorrect factorization can lead to significant errors in subsequent calculations or applications of the solution. Therefore, it is crucial to ensure the accuracy of each step in the factoring process.
Option B, -7(x - 5)(x + 2), is also incorrect. Although it has a similar structure to the correct answer, the factors (x - 5) and (x + 2) do not combine to produce the middle term of -4x in the original trinomial. This underscores the importance of precise factorization. Even a small difference in the factors can result in a completely different expression when expanded. Attention to detail is paramount in factoring, as even minor errors can lead to incorrect results.
Option D, (-7x + 10)(x - 2), is incorrect as well. While it has some elements similar to the correct factorization, the signs within the factors are incorrect. When expanded, this option would yield a trinomial with a different middle term and constant term than the original expression. This emphasizes the significance of correctly identifying the signs in each factor. Sign errors are a common pitfall in factoring, and careful attention to detail can help avoid them.
In conclusion, the correct factorization of the trinomial -7x² - 4x + 20 is -1(7x - 10)(x + 2), which corresponds to option C. This process involved factoring out a -1, identifying key factors, rewriting the middle term, factoring by grouping, and careful attention to signs. Factoring trinomials is a fundamental skill in algebra, and mastering it requires practice and a clear understanding of the underlying principles. By following these steps and practicing regularly, you can confidently tackle a wide variety of factoring problems.
Factoring is not just a mathematical technique; it is a way of thinking about algebraic expressions. It involves breaking down complex expressions into simpler components, which can then be manipulated more easily. This skill is essential for solving equations, simplifying expressions, and understanding the relationships between different mathematical concepts. By mastering factoring, you will gain a deeper appreciation for the structure of algebra and enhance your problem-solving abilities in mathematics and related fields.
The ability to factor trinomials efficiently and accurately is a cornerstone of algebraic proficiency. It is a skill that is frequently used in higher-level mathematics courses, such as calculus and linear algebra. Furthermore, factoring concepts extend beyond mathematics and find applications in various fields, including physics, engineering, and computer science. Therefore, investing time and effort in mastering factoring is a worthwhile endeavor that will pay dividends throughout your academic and professional career.
The journey of factoring the trinomial -7x² - 4x + 20 has been a comprehensive exploration of algebraic techniques and problem-solving strategies. We have not only arrived at the correct factorization but also delved into the underlying principles and the importance of careful attention to detail. This approach is crucial for developing a deep understanding of mathematics and for building the confidence to tackle complex problems. By continuing to practice and refine your skills, you will become a proficient problem solver and a confident mathematician.
