Determining The Coefficient B In Factored Trinomials

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In this article, we will delve into the process of determining the coefficient B in the factored form of a given trinomial. Specifically, we will focus on trinomials expressed in the form $ax^2 + bxy + cy^2$, where a, b, and c are constants. Factoring trinomials is a fundamental skill in algebra, with applications in various mathematical fields and real-world problem-solving scenarios. Understanding how to factor trinomials and identify coefficients like B is crucial for success in algebra and beyond. We will explore the underlying principles of factoring, demonstrate the step-by-step process involved, and provide examples to solidify your understanding. Whether you are a student learning about factoring for the first time or someone looking to refresh your skills, this guide will provide you with the knowledge and confidence to tackle these types of problems.

Understanding Trinomial Factoring

Trinomial factoring involves breaking down a trinomial expression into the product of two binomials. This process relies on the distributive property and the relationship between the coefficients of the trinomial and the constants in the binomial factors. For a trinomial in the form $ax^2 + bxy + cy^2$, the factored form will generally look like $(px + qy)(rx + sy)$, where p, q, r, and s are constants. The goal of factoring is to find the values of these constants that, when the binomials are multiplied, result in the original trinomial expression. This often involves considering the factors of the leading coefficient (a) and the constant term (c), and then finding a combination of factors that, when combined, yields the middle coefficient (b). The sign of the middle term and the constant term can provide clues about the signs of the constants in the binomial factors. For instance, if the constant term is positive and the middle term is also positive, both constants in the binomial factors will likely be positive. If the constant term is positive but the middle term is negative, both constants in the binomial factors will likely be negative. However, if the constant term is negative, one constant in the binomial factors will be positive, and the other will be negative. By carefully analyzing the signs and factors of the coefficients in the trinomial, we can systematically determine the appropriate constants for the binomial factors and arrive at the factored form of the trinomial.

Problem Statement

Let's consider the trinomial:

2x2+14xy−36y2=2(x+By)(x−2y)2x^2 + 14xy - 36y^2 = 2(x + By)(x - 2y)

Our objective is to determine the value of the coefficient B in the factored form of this trinomial. This problem requires us to apply our understanding of trinomial factoring and the distributive property to identify the correct value of B that satisfies the given equation. The first step in solving this problem is to recognize that the trinomial has a common factor of 2, which has already been factored out on the right-hand side of the equation. This simplifies the factoring process, as we can focus on factoring the remaining trinomial expression within the parentheses. The factored form presented on the right-hand side provides a significant clue about the structure of the factors, indicating that one of the binomial factors is $(x - 2y)$. This information allows us to strategically approach the factoring process by focusing on identifying the second binomial factor that, when multiplied by $(x - 2y)$, yields the original trinomial expression (after accounting for the common factor of 2). By carefully considering the coefficients and signs in the trinomial and the known binomial factor, we can deduce the value of B that makes the equation true.

Step-by-Step Solution

  1. Factor out the common factor:

Begin by factoring out the common factor of 2 from the trinomial:

2(x2+7xy−18y2)=2(x+By)(x−2y)2(x^2 + 7xy - 18y^2) = 2(x + By)(x - 2y)

This step simplifies the expression and allows us to focus on factoring the remaining trinomial $x^2 + 7xy - 18y^2$. Factoring out common factors is a crucial first step in many factoring problems, as it can significantly reduce the complexity of the expression and make the subsequent factoring steps easier to manage. By removing the common factor of 2, we are left with a simpler trinomial that has smaller coefficients, making it easier to identify the appropriate factors. This step also helps to align the problem with the given factored form, making it clearer how to determine the value of B.

  1. Focus on the trinomial:

Now, we need to factor the trinomial $x^2 + 7xy - 18y^2$. We are looking for two binomials of the form $(x + qy)(x + sy)$ such that when multiplied, they result in the given trinomial. To find these binomials, we need to identify two numbers, q and s, that multiply to -18 (the constant term) and add up to 7 (the coefficient of the xy term). This process involves considering the factors of -18 and systematically testing different combinations to find the pair that satisfies the given conditions. The factors of -18 include pairs such as (1, -18), (-1, 18), (2, -9), (-2, 9), (3, -6), and (-3, 6). By examining these pairs, we can identify the pair that adds up to 7, which will give us the values of q and s needed to construct the binomial factors.

  1. Identify the factors:

We need two numbers that multiply to -18 and add to 7. The numbers 9 and -2 satisfy these conditions:

  • 9 × (-2) = -18
  • 9 + (-2) = 7

Identifying the correct factors is a crucial step in the factoring process. It requires a combination of understanding the relationships between the coefficients in the trinomial and systematically testing different factor pairs. In this case, the negative constant term (-18) indicates that one of the factors must be positive and the other must be negative. The positive coefficient of the xy term (7) suggests that the larger factor should be positive. By considering these clues and examining the factor pairs of -18, we can efficiently identify the correct pair (9 and -2) that satisfies both conditions.

  1. Write the factored form:

Therefore, the trinomial can be factored as:

x2+7xy−18y2=(x+9y)(x−2y)x^2 + 7xy - 18y^2 = (x + 9y)(x - 2y)

Once the factors have been identified, writing the factored form is a straightforward process. We simply substitute the values of q and s that we found in the previous step into the binomial form $(x + qy)(x + sy)$. In this case, since q = 9 and s = -2, the factored form becomes $(x + 9y)(x - 2y)$. This factored form represents the product of two binomials that, when multiplied, will result in the original trinomial expression. It is important to double-check the factored form by multiplying the binomials to ensure that it matches the original trinomial. This step helps to verify the correctness of the factoring process and prevent errors.

  1. Substitute back into the original equation:

Substitute this back into the original equation:

2(x+9y)(x−2y)=2(x+By)(x−2y)2(x + 9y)(x - 2y) = 2(x + By)(x - 2y)

Substituting the factored form back into the original equation allows us to directly compare the factored expression with the given factored form and determine the value of B. This step is crucial for solving the problem, as it establishes a clear relationship between the factored trinomial and the unknown coefficient B. By replacing the trinomial with its factored form, we can see that the equation now has the same binomial factor $(x - 2y)$ on both sides, which simplifies the process of identifying the corresponding coefficient in the other binomial factor.

  1. Determine the value of B:

By comparing the two sides of the equation, we can see that:

B=9B = 9

By comparing the two sides of the equation after substitution, we can directly identify the value of B. Since the binomial factor $(x - 2y)$ is the same on both sides, we can focus on the other binomial factor to determine the value of B. In this case, the binomial factor on the left-hand side is $(x + 9y)$, and the binomial factor on the right-hand side is $(x + By)$. By comparing the coefficients of the y term in these two binomials, we can see that B must be equal to 9. This is the final step in the solution process, and it provides the answer to the problem statement.

Final Answer

Therefore, the value of the coefficient B in the factored form of the given trinomial is 9.

This step-by-step solution demonstrates the process of factoring a trinomial and identifying a specific coefficient in its factored form. By following this approach, you can confidently solve similar problems and deepen your understanding of trinomial factoring. The key to success in these types of problems lies in carefully analyzing the coefficients and signs in the trinomial, identifying the appropriate factors, and systematically working through the steps to arrive at the final answer. Practice and familiarity with factoring techniques will help you develop the skills needed to tackle more complex factoring problems in algebra and beyond.

Additional Practice Problems

To further enhance your understanding of trinomial factoring and coefficient determination, consider working through these additional practice problems:

  1. Factor the trinomial $3x^2 - 10xy - 8y^2$ and identify the coefficient of y in the factored form.
  2. Given the equation $4x^2 + 20xy + 25y^2 = (Ax + 5y)^2$, find the value of the coefficient A.
  3. Determine the value of C in the factored form of the trinomial $x^2 - 5xy + 6y^2 = (x - 2y)(x + Cy)$.

By tackling these practice problems, you can reinforce the concepts and techniques discussed in this article and develop your problem-solving skills in algebra. Remember to break down each problem into smaller steps, carefully analyze the coefficients and signs, and apply the appropriate factoring strategies. With practice and persistence, you will become more proficient in factoring trinomials and identifying coefficients in their factored forms.