Factor The Trinomial 5z^2 - 7z - 6. Express The Factored Form As (5z + [ ] )(z - [ ]).
Factoring trinomials can often feel like solving a puzzle, especially when the leading coefficient is not 1. In this comprehensive guide, we will thoroughly explore the process of factoring the trinomial 5z² - 7z - 6. This step-by-step approach, designed to clarify the methods used, will transform a potentially challenging task into a manageable one. Whether you're a student grappling with algebra or a math enthusiast eager to expand your skills, this detailed breakdown will illuminate the techniques needed to successfully factor such expressions. Let's embark on this mathematical journey together, unlocking the secrets hidden within this quadratic equation.
Understanding the Problem
Before diving into the solution, it’s crucial to fully understand the problem we're tackling. We are given the quadratic trinomial 5z² - 7z - 6 and tasked with expressing it as a product of two binomials. This process, known as factoring the trinomial, essentially reverses the multiplication process, allowing us to decompose a complex expression into simpler components. Recognizing the structure of the trinomial is the first step towards finding its factors. A trinomial, by definition, consists of three terms, and in this case, we have a quadratic trinomial because the highest power of the variable z is 2. The general form of a quadratic trinomial is ax² + bx + c, where a, b, and c are constants. In our specific case, a = 5, b = -7, and c = -6. These coefficients play a critical role in determining the factors, guiding our strategy as we seek the binomials that, when multiplied, will yield the original trinomial. Grasping this foundational understanding is paramount to a successful factoring process. Let's proceed to explore the methods we can use to break down this expression.
Methods for Factoring Trinomials
Several methods can be employed when factoring the trinomial, each with its own strengths and suitability depending on the specific problem. Two popular methods include the trial and error method and the AC method. The trial and error method, as its name suggests, involves making educated guesses and checking if the binomial factors multiply to give the original trinomial. While this method can be quick for simpler trinomials, it can become time-consuming and less efficient for more complex expressions with larger coefficients or negative signs. On the other hand, the AC method provides a more systematic approach, reducing the guesswork involved. This method relies on finding two numbers that multiply to ac (the product of the leading coefficient and the constant term) and add up to b (the coefficient of the linear term). By decomposing the middle term using these numbers, we can then factor by grouping, a technique that breaks the trinomial into pairs of terms that share common factors. This method is particularly useful when dealing with trinomials that have a leading coefficient other than 1, as in our case with 5z² - 7z - 6. Let's delve deeper into the AC method, which will be our primary tool for factoring the trinomial in question.
Applying the AC Method to 5z² - 7z - 6
The AC method provides a structured approach to factoring the trinomial 5z² - 7z - 6. This method hinges on identifying two key numbers that satisfy specific conditions derived from the coefficients of the trinomial. To begin, we first determine the product of a and c, where a is the coefficient of the z² term and c is the constant term. In our case, a = 5 and c = -6, so ac = 5 * (-6) = -30. Next, we need to find two numbers that multiply to -30 and add up to b, which is the coefficient of the z term. Here, b = -7. Through careful consideration, we identify the numbers -10 and 3 as the pair that meets these criteria: (-10) * 3 = -30 and (-10) + 3 = -7. These numbers are the key to rewriting the middle term of the trinomial. We replace the term -7z with -10z + 3z, effectively splitting the middle term into two parts. This transformation yields a new expression: 5z² - 10z + 3z - 6. This rewritten expression is equivalent to the original trinomial but is now structured in a way that allows us to factor by grouping. Factoring by grouping involves pairing the terms and extracting common factors from each pair, a technique that will bring us closer to the factored form of the trinomial.
Factoring by Grouping
Having rewritten the trinomial as 5z² - 10z + 3z - 6, we now proceed with factoring by grouping. This technique involves pairing the first two terms and the last two terms, and then extracting the greatest common factor (GCF) from each pair. Looking at the first pair, 5z² - 10z, we identify the GCF as 5z. Factoring out 5z from this pair, we get 5z(z - 2). Next, we consider the second pair, 3z - 6. The GCF here is 3, and factoring it out gives us 3(z - 2). Now, we have the expression 5z(z - 2) + 3(z - 2). Notice that both terms share a common binomial factor, (z - 2). This is a crucial observation, as it allows us to factor out the binomial (z - 2) from the entire expression. When we factor out (z - 2), we are left with 5z from the first term and 3 from the second term, resulting in (z - 2)(5z + 3). This factored form represents the original trinomial 5z² - 7z - 6 expressed as a product of two binomials. The process of factoring by grouping not only simplifies the expression but also provides a clear pathway to the final factored form, showcasing the elegance of algebraic manipulation. We have successfully factored the trinomial using the AC method and factoring by grouping.
The Final Factored Form
After meticulously applying the AC method and factoring by grouping, we have arrived at the final factored form of the trinomial 5z² - 7z - 6. The expression has been successfully decomposed into the product of two binomials: (5z + 3)(z - 2). This represents the culmination of our step-by-step process, where we started by understanding the problem, identifying the appropriate method, and executing the factoring techniques. To ensure the accuracy of our result, it’s always a good practice to verify the factored form by expanding it. Expanding (5z + 3)(z - 2) using the distributive property (also known as the FOIL method) should yield the original trinomial. Let's perform this check:
(5z + 3)(z - 2) = 5z(z) + 5z(-2) + 3(z) + 3(-2) = 5z² - 10z + 3z - 6 = 5z² - 7z - 6.
The expansion confirms that our factored form is correct. We have successfully factored the trinomial 5z² - 7z - 6 into (5z + 3)(z - 2). This achievement not only demonstrates the power of algebraic techniques but also solidifies our understanding of how to manipulate and simplify complex expressions. The factored form offers valuable insights into the roots and behavior of the quadratic equation, making it an essential tool in various mathematical applications.
Conclusion
In this comprehensive guide, we have successfully factored the trinomial 5z² - 7z - 6 into its factored form, (5z + 3)(z - 2). Through a detailed exploration of the AC method and factoring by grouping, we have demonstrated a systematic approach to tackling this type of problem. We began by understanding the structure of the trinomial and the importance of its coefficients. Then, we applied the AC method to identify the key numbers needed to rewrite the middle term. This led us to the technique of factoring by grouping, where we extracted common factors to simplify the expression. Finally, we verified our result by expanding the factored form, ensuring its accuracy. This journey through the factoring process has not only provided us with the solution to this specific problem but has also equipped us with a valuable toolkit for factoring other trinomials. The ability to factor trinomials is a fundamental skill in algebra, with applications ranging from solving equations to simplifying complex expressions. By mastering these techniques, we can confidently approach mathematical challenges and unlock the underlying structure of quadratic expressions. Factoring may initially seem like a daunting task, but with a methodical approach and a solid understanding of the underlying principles, it becomes a manageable and even elegant process.