Calculate The Value Of 'a' If The Independent Term Of P(x) = (x + A)(x + 4) Is 28.

Understanding the Problem: Independent Term in a Polynomial

In the realm of mathematics, specifically within the study of polynomials, the concept of an independent term holds significant importance. When we talk about a polynomial, we are essentially dealing with an expression comprising variables and coefficients, combined through addition, subtraction, and multiplication, with non-negative integer exponents. The independent term, also referred to as the constant term, is the one that doesn't have any variable attached to it. It's the term that remains constant regardless of the value of the variable. Finding the independent term is crucial in various mathematical applications, including solving equations, graphing functions, and understanding the behavior of polynomials. This exploration aims to delve deeper into the process of calculating the value of a specific variable within a polynomial expression, given that the independent term is known. Through this examination, readers will gain a comprehensive understanding of the underlying principles and methodologies involved in solving such problems, thereby enhancing their mathematical proficiency and problem-solving skills. A clear grasp of these concepts is vital for anyone pursuing studies in algebra, calculus, and related fields, where polynomials play a foundational role. By focusing on the interplay between variables, coefficients, and the independent term, we can unravel complex mathematical relationships and arrive at precise solutions.

Problem Statement: P(x) = (x + a)(x + 4) and the Independent Term

Our specific challenge is to determine the value of 'a' in the polynomial P(x) = (x + a)(x + 4), given that the independent term of this polynomial is 28. This problem combines the concepts of polynomial expansion and the identification of the independent term. The first step involves expanding the given polynomial expression. Expanding a polynomial means multiplying out the factors to get a standard form polynomial, which is a sum of terms, each consisting of a coefficient and a power of the variable. In this case, we need to multiply (x + a) by (x + 4). This process utilizes the distributive property of multiplication over addition, ensuring that each term in the first factor is multiplied by each term in the second factor. Once we've expanded the polynomial, we can then identify the independent term. The independent term is the one that doesn't contain the variable 'x'. In other words, it's the constant term in the expanded form of the polynomial. The problem states that this independent term is equal to 28. This piece of information is the key to finding the value of 'a'. By setting the independent term equal to 28, we create an equation. Solving this equation for 'a' will give us the value we're looking for. This task requires careful algebraic manipulation and a solid understanding of the properties of polynomials. Successfully navigating this problem not only provides the solution for 'a' but also reinforces the fundamental skills necessary for tackling more complex algebraic challenges.

Step-by-Step Solution: Expanding the Polynomial and Finding the Independent Term

To solve this problem, we will proceed with a step-by-step approach. First, let's expand the polynomial P(x) = (x + a)(x + 4). Using the distributive property, we multiply each term in the first parenthesis by each term in the second parenthesis: P(x) = x(x + 4) + a(x + 4). This expands to P(x) = x^2 + 4x + ax + 4a. Now, we can rearrange the terms to group like terms together: P(x) = x^2 + (4 + a)x + 4a. In this expanded form, it becomes clear that the independent term is 4a, as it is the term without any 'x' variable attached to it. The problem states that the independent term is 28, so we set 4a equal to 28: 4a = 28. To solve for 'a', we divide both sides of the equation by 4: a = 28 / 4. This gives us a = 7. Therefore, the value of 'a' that makes the independent term of the polynomial P(x) equal to 28 is 7. This methodical approach of expanding the polynomial, identifying the independent term, and setting up an equation allows us to systematically find the solution. This process not only provides the answer but also reinforces the importance of algebraic manipulation and equation-solving skills in mathematics. By carefully following these steps, we can confidently tackle similar problems involving polynomials and independent terms.

Verifying the Solution: Substituting 'a' Back into the Polynomial

To ensure the accuracy of our solution, it's crucial to verify that the value we found for 'a' indeed results in an independent term of 28. We determined that a = 7. Now, let's substitute this value back into the original polynomial: P(x) = (x + a)(x + 4). Replacing 'a' with 7, we get: P(x) = (x + 7)(x + 4). Next, we expand this polynomial using the distributive property, just as we did before: P(x) = x(x + 4) + 7(x + 4). This expands to P(x) = x^2 + 4x + 7x + 28. Combining like terms, we have: P(x) = x^2 + 11x + 28. Now, we can clearly see the independent term, which is the constant term without any 'x' variable. In this case, the independent term is 28, which matches the value given in the problem statement. This verification confirms that our solution a = 7 is correct. By substituting the value back into the original equation and checking that it satisfies the given condition, we gain confidence in our answer and ensure that we haven't made any algebraic errors along the way. This process of verification is an essential step in problem-solving, particularly in mathematics, as it helps to identify and correct any mistakes, leading to a more accurate and reliable solution. It also reinforces the understanding of the underlying mathematical concepts and principles involved in the problem.

Conclusion: The Value of 'a' and Its Significance

In conclusion, by meticulously expanding the polynomial P(x) = (x + a)(x + 4), identifying the independent term, and solving the resulting equation, we have successfully determined that the value of a is 7. This solution satisfies the condition that the independent term of the polynomial is 28. The significance of this problem extends beyond simply finding the value of 'a'. It highlights the fundamental concepts of polynomial algebra, including expansion, the identification of terms, and equation solving. Understanding these concepts is crucial for success in more advanced mathematical topics, such as calculus and differential equations, where polynomials play a central role. Moreover, the process of verifying the solution by substituting the value back into the original equation reinforces the importance of accuracy and attention to detail in mathematical problem-solving. This meticulous approach ensures that the solution not only satisfies the given conditions but also aligns with the underlying mathematical principles. The ability to confidently manipulate polynomials and solve related problems is a valuable skill in various fields, including engineering, physics, and computer science. By mastering these foundational concepts, students and professionals alike can tackle complex problems and develop a deeper understanding of the mathematical world. The determination of 'a' in this specific case serves as a stepping stone towards a broader appreciation of the power and versatility of polynomial algebra in both theoretical and practical applications.