Simplifying Algebraic Expressions A Step By Step Guide
Understanding the Problem
In this mathematical problem, we are tasked with performing the indicated operations and simplifying the expression (t+7)(t+8)-9(t+4). This involves expanding the products, distributing the constants, and then combining like terms to arrive at a simplified form. Mastering such algebraic manipulations is crucial for success in various areas of mathematics, including calculus, linear algebra, and differential equations. The ability to simplify expressions efficiently and accurately is a cornerstone of problem-solving in these fields. This expression combines polynomial multiplication with the distributive property, two fundamental concepts in algebra. By meticulously applying these concepts, we can break down the expression into manageable components, ultimately leading to a simplified result. The process highlights the importance of paying attention to detail and ensuring that each step is performed correctly. Simplifying algebraic expressions is not just an academic exercise; it has practical applications in various scientific and engineering disciplines, where complex equations need to be manipulated to find solutions or to gain insights into the behavior of systems. For instance, in physics, simplifying equations can help in understanding the motion of objects, the behavior of circuits, or the properties of electromagnetic waves. In engineering, it can be used to design structures, optimize processes, or analyze the stability of systems. Therefore, a solid understanding of algebraic simplification is an invaluable skill for anyone pursuing a career in these fields. Moreover, the problem-solving skills developed through algebraic manipulation extend beyond mathematics and science. The ability to break down complex problems into smaller, more manageable parts, to identify patterns, and to apply logical reasoning are all essential skills in everyday life and in various professional settings. Therefore, mastering this type of problem not only enhances mathematical proficiency but also contributes to the development of valuable cognitive abilities.
Step-by-Step Solution
To simplify the expression (t+7)(t+8)-9(t+4), we'll proceed step-by-step, carefully applying the distributive property and combining like terms. First, we need to expand the product of the two binomials, (t+7)(t+8). This can be done using the FOIL method (First, Outer, Inner, Last) or by directly applying the distributive property. Both methods yield the same result, but the FOIL method is a commonly used mnemonic that helps to ensure all terms are multiplied correctly. Applying the FOIL method, we have: First: t * t = t^2, Outer: t * 8 = 8t, Inner: 7 * t = 7t, Last: 7 * 8 = 56. Combining these terms, we get t^2 + 8t + 7t + 56. Next, we can combine the like terms 8t and 7t, which gives us 15t. So, the expanded form of (t+7)(t+8) is t^2 + 15t + 56. Now, let's move on to the second part of the expression, -9(t+4). Here, we need to distribute the -9 across both terms inside the parentheses. Multiplying -9 by t gives -9t, and multiplying -9 by 4 gives -36. So, -9(t+4) expands to -9t - 36. Now we have expanded both parts of the expression. The original expression (t+7)(t+8)-9(t+4) is now t^2 + 15t + 56 - 9t - 36. The next step is to combine like terms. We have the t^2 term, the t terms (15t and -9t), and the constant terms (56 and -36). Combining the t terms, 15t - 9t equals 6t. Combining the constant terms, 56 - 36 equals 20. So, after combining like terms, the expression simplifies to t^2 + 6t + 20. This is the simplified form of the original expression. We have expanded the products, distributed the constants, and combined like terms to arrive at this result. Each step in the process is crucial, and accuracy is essential to avoid errors. By following this step-by-step approach, we can confidently simplify complex algebraic expressions. The final result, t^2 + 6t + 20, is a quadratic expression in standard form, where the terms are arranged in descending order of their exponents. This simplified form is easier to work with in many contexts, such as solving equations or graphing functions. The process of simplifying expressions is a fundamental skill in algebra, and mastering it opens the door to more advanced mathematical concepts.
Detailed Breakdown of Steps
Let's delve deeper into the steps required to simplify the expression (t+7)(t+8)-9(t+4). Each step is a building block, and understanding the rationale behind each action is crucial for mastering algebraic simplification. The first key step is expanding the product of the binomials (t+7)(t+8). As mentioned earlier, this can be achieved using the FOIL method or the distributive property. The FOIL method, which stands for First, Outer, Inner, Last, provides a systematic way to ensure that each term in the first binomial is multiplied by each term in the second binomial. 'First' refers to multiplying the first terms of each binomial (t * t). 'Outer' refers to multiplying the outer terms (t * 8). 'Inner' refers to multiplying the inner terms (7 * t), and 'Last' refers to multiplying the last terms (7 * 8). By following this order, we can avoid missing any terms. Alternatively, we can use the distributive property, which states that a(b + c) = ab + ac. In this case, we can think of (t+7)(t+8) as (t+7) multiplied by (t+8). So, we distribute the (t+7) across the terms of (t+8): (t+7) * t + (t+7) * 8. Then, we further distribute the t and 8 across the terms of (t+7): t * t + 7 * t + 8 * t + 7 * 8, which is equivalent to the FOIL method. Either way, expanding (t+7)(t+8) results in t^2 + 8t + 7t + 56. The next step is to simplify this expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, 8t and 7t are like terms because they both have the variable t raised to the power of 1. We can combine them by adding their coefficients: 8t + 7t = 15t. So, the expanded and simplified form of (t+7)(t+8) is t^2 + 15t + 56. The second part of the original expression is -9(t+4). Here, we need to distribute the -9 across the terms inside the parentheses. This means multiplying -9 by both t and 4. When we multiply -9 by t, we get -9t. When we multiply -9 by 4, we get -36. So, -9(t+4) expands to -9t - 36. It's crucial to remember to distribute the negative sign along with the 9. Now that we have expanded both parts of the original expression, we can write the entire expression as t^2 + 15t + 56 - 9t - 36. The final step is to combine all like terms. We have one term with t^2 (t^2), two terms with t (15t and -9t), and two constant terms (56 and -36). Combining the t terms, 15t - 9t equals 6t. Combining the constant terms, 56 - 36 equals 20. Therefore, the simplified expression is t^2 + 6t + 20. This is a quadratic expression in standard form, where the terms are arranged in descending order of their exponents.
Common Mistakes to Avoid
When simplifying algebraic expressions like (t+7)(t+8)-9(t+4), it's easy to make mistakes if one is not careful. Recognizing and avoiding these common pitfalls can significantly improve accuracy and confidence in problem-solving. One of the most common errors occurs during the expansion of binomial products, such as (t+7)(t+8). A mistake can easily happen if the FOIL method or the distributive property is not applied correctly. For example, students may forget to multiply all terms or may incorrectly multiply the coefficients. It's crucial to systematically multiply each term in the first binomial by each term in the second binomial, paying close attention to the signs. Another common mistake involves distributing the constant in expressions like -9(t+4). It's essential to remember to distribute the negative sign along with the number. Failing to do so can lead to incorrect signs in the expanded expression. For example, multiplying -9 by t should result in -9t, and multiplying -9 by 4 should result in -36. A mistake occurs if the student only multiplies 9 by t and 4, resulting in 9t + 36, and then subtracts this entire expression, which is an incorrect application of the distributive property. Another frequent error occurs when combining like terms. Like terms are terms that have the same variable raised to the same power. For example, 15t and -9t are like terms, but 15t and 56 are not. When combining like terms, it's important to only add or subtract the coefficients of the terms that are alike. A mistake can arise if terms with different variables or exponents are combined, which would lead to an incorrect simplification. For instance, attempting to combine t^2 with 6t or 20 would be an error. Additionally, students sometimes make mistakes when dealing with the order of operations. It's crucial to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This means that any operations inside parentheses should be performed first, followed by exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). In this specific problem, the multiplication (expanding the binomials and distributing the constant) should be done before combining like terms. Another common mistake is rushing through the steps and not paying attention to detail. Algebraic simplification requires careful attention to each step, and it's important to double-check the work to ensure accuracy. Skipping steps or making assumptions can lead to errors. It's always a good idea to write out each step clearly and methodically to minimize the chances of making a mistake. Finally, students sometimes make errors when rewriting the expression. When moving terms around or combining like terms, it's crucial to rewrite the expression accurately. A mistake in rewriting can lead to an incorrect final answer, even if all the individual steps were performed correctly.
Practice Problems
To solidify your understanding of simplifying algebraic expressions, working through practice problems is invaluable. These problems provide an opportunity to apply the concepts learned and to identify any areas that may need further attention. Let's explore a few practice problems similar to (t+7)(t+8)-9(t+4).
Practice Problem 1: Simplify the expression (x+3)(x+5)-2(x+1). This problem is structured similarly to the original problem. First, expand the product of the binomials (x+3)(x+5) using the FOIL method or the distributive property. This gives you x^2 + 5x + 3x + 15. Then, combine like terms to get x^2 + 8x + 15. Next, distribute the -2 in -2(x+1), which results in -2x - 2. Finally, combine all like terms in the expression x^2 + 8x + 15 - 2x - 2. This simplifies to x^2 + 6x + 13.
Practice Problem 2: Simplify the expression (y-4)(y+2)+3(y-1). This problem introduces negative numbers, which can sometimes be a source of errors. Expand (y-4)(y+2) to get y^2 + 2y - 4y - 8. Combine like terms to get y^2 - 2y - 8. Distribute the 3 in 3(y-1) to get 3y - 3. Combine all like terms in the expression y^2 - 2y - 8 + 3y - 3. This simplifies to y^2 + y - 11.
Practice Problem 3: Simplify the expression (2z+1)(z-3)-4(z+2). This problem includes a coefficient in front of the variable, which adds another layer of complexity. Expand (2z+1)(z-3) to get 2z^2 - 6z + z - 3. Combine like terms to get 2z^2 - 5z - 3. Distribute the -4 in -4(z+2) to get -4z - 8. Combine all like terms in the expression 2z^2 - 5z - 3 - 4z - 8. This simplifies to 2z^2 - 9z - 11.
Practice Problem 4: Simplify the expression (a-2)(a-5)+6(a+3). This problem provides additional practice with negative numbers and distribution. Expand (a-2)(a-5) to get a^2 - 5a - 2a + 10. Combine like terms to get a^2 - 7a + 10. Distribute the 6 in 6(a+3) to get 6a + 18. Combine all like terms in the expression a^2 - 7a + 10 + 6a + 18. This simplifies to a^2 - a + 28.
Practice Problem 5: Simplify the expression (3b+2)(b-1)-2(b-4). This problem includes a coefficient in front of the variable in both binomials. Expand (3b+2)(b-1) to get 3b^2 - 3b + 2b - 2. Combine like terms to get 3b^2 - b - 2. Distribute the -2 in -2(b-4) to get -2b + 8. Combine all like terms in the expression 3b^2 - b - 2 - 2b + 8. This simplifies to 3b^2 - 3b + 6. By working through these practice problems, you can gain confidence in your ability to simplify algebraic expressions. Remember to take your time, pay attention to detail, and double-check your work. With practice, simplifying algebraic expressions will become a routine task.
Conclusion
In conclusion, simplifying the expression (t+7)(t+8)-9(t+4) involves a series of steps that require careful attention to detail and a solid understanding of algebraic principles. We began by expanding the product of the binomials (t+7)(t+8), which yielded t^2 + 15t + 56. Then, we distributed the -9 across the terms in (t+4), resulting in -9t - 36. The final step was to combine like terms, which led us to the simplified expression t^2 + 6t + 20. This process highlights the importance of mastering fundamental algebraic techniques, such as the distributive property, the FOIL method, and the combination of like terms. These skills are not only essential for success in mathematics but also have broad applications in various scientific and engineering disciplines. By systematically applying these techniques, we can break down complex expressions into simpler, more manageable forms, making them easier to work with and interpret. The ability to simplify expressions efficiently and accurately is a cornerstone of problem-solving in many fields. Moreover, the problem-solving skills developed through algebraic manipulation extend beyond mathematics. The ability to break down complex problems into smaller, more manageable parts, to identify patterns, and to apply logical reasoning are all valuable skills in everyday life and in various professional settings. Therefore, mastering this type of problem not only enhances mathematical proficiency but also contributes to the development of valuable cognitive abilities. Through practice and a thorough understanding of the underlying principles, one can become proficient in simplifying algebraic expressions and confidently tackle more complex mathematical problems. Remember to pay attention to detail, avoid common mistakes, and always double-check your work to ensure accuracy. With consistent effort and a systematic approach, algebraic simplification can become a routine and even enjoyable part of your mathematical journey. The simplified expression, t^2 + 6t + 20, represents a more concise and easily understandable form of the original expression, making it readily applicable in further mathematical operations or analyses. This process underscores the power and elegance of algebraic simplification in transforming complex expressions into simpler, more manageable forms, thereby facilitating problem-solving and fostering a deeper understanding of mathematical concepts. As you continue to practice and refine your skills in algebraic manipulation, you'll find that you're better equipped to tackle a wide range of mathematical challenges, both in academic settings and in real-world applications. So, embrace the challenge, practice diligently, and enjoy the satisfaction of mastering this essential mathematical skill.
