Polynomial Puzzle: Finding The Missing Addend

Hey math enthusiasts! Let's dive into a cool algebra problem. We're given a sum of two polynomials, and one of the addends. Our mission? To find the other addend! It's like a mathematical detective story, and we're the solvers. Let's break it down step by step and crack this polynomial puzzle. So, the question tells us that the sum of two polynomials is 10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2. One of the addends is -5a^2b^2 + 12a^2b - 5. Our goal is to determine the other addend. This is a classic example of polynomial arithmetic, and with a little bit of algebraic know-how, we'll nail it. Ready to roll up your sleeves and get started? Let's go!

Understanding the Problem

Alright, before we jump into the calculations, let's make sure we're all on the same page. The problem is all about polynomials, which are expressions made up of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. In this case, we're dealing with polynomials in terms of a and b. The sum of two polynomials means we've added them together, and the result is the given polynomial. We are provided with both a sum and one addend. Our task is to identify the other one. Think of it like a missing piece in a puzzle. To find the missing addend, we need to reverse the addition process. We will do this by subtracting the known addend from the sum to find the unknown addend. This is similar to solving an equation where we isolate the unknown variable. Knowing this crucial step helps us to accurately calculate the unknown polynomial. This step-by-step approach simplifies the overall process.

Breaking Down the Math

So, to get the other addend, we're going to subtract the known addend from the total sum. Here's how it looks:

Other Addend = (Sum) - (Known Addend)

Let's plug in our polynomials:

Other Addend = (10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2) - (-5a^2b^2 + 12a^2b - 5)

Notice the subtraction of the second polynomial. Remember, subtracting a polynomial means subtracting each of its terms. This is where it's important to be careful with signs. The subtraction of the negative terms in the second polynomial is critical. It involves changing the sign of each term inside the parentheses. So, let's change the signs correctly! This is where we start to actually see the value in this mathematical problem and begin to understand its process.

Solving for the Other Addend

Okay, let's carry out the subtraction step by step. We'll pay close attention to the signs to avoid any mix-ups. We will be combining like terms here. Like terms are terms that have the same variables raised to the same powers. For instance, 3x^2 and 5x^2 are like terms, while 3x^2 and 3x are not. Here we are doing the same with polynomials. It is a critical aspect of simplifying the expression and getting to the answer. Let's start the process:

10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2 - (-5a^2b^2 + 12a^2b - 5)

Distribute the negative sign (subtracting the second polynomial):

10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2 + 5a^2b^2 - 12a^2b + 5

Now, combine like terms. This means we add or subtract the coefficients of terms with the same variables and exponents:

• For a^2b^2 terms: 10a^2b^2 + 5a^2b^2 = 15a^2b^2
• For a^2b terms: -8a^2b - 12a^2b = -20a^2b
• The 6ab^2 and -4ab terms remain as they are because there are no other like terms.
• For constant terms: 2 + 5 = 7

Putting it all together, we get:

15a^2b^2 - 20a^2b + 6ab^2 - 4ab + 7

This is the other addend! We've successfully navigated the subtraction of the polynomials, correctly identified and combined like terms, and have found the solution. High five! Now we know what our other addend is. Let's proceed to the answer options and verify our findings.

Verifying the Answer

Now that we have the other addend, let's check it against the answer options provided to ensure we have the correct response. Remember, we calculated the other addend to be 15a^2b^2 - 20a^2b + 6ab^2 - 4ab + 7. Let's see which option matches this calculation. The aim here is to match our work to the options and ensure we have done everything correctly. This step is critical to the exam and ensuring the highest possible mark. Always do a double check.

• Option A: 15a^2b^2 - 20a^2b + 6ab^2 - 4ab + 7. Bingo! This is a perfect match to our calculated addend.
• Option B: 5a^2b^2 - 20a^2b^2 + 7. This is incorrect because the coefficients of the polynomial do not match our calculations.
• Option C: 5a^2b^2 + 4a^2b^2 + 6ab - 4ab - 3. This does not match our calculated addend.

Therefore, based on our calculations and comparison with the answer choices, the correct answer is Option A. That’s how we successfully solved this polynomial problem, by subtracting the known addend from the total sum, paying close attention to the signs, and then identifying the correct option that matches our calculations.

Conclusion: Mastering Polynomial Arithmetic

And that's a wrap, guys! We have successfully found the missing addend in this polynomial puzzle. We started by understanding the concept of polynomial addition and subtraction, carefully applied the subtraction operation, paying close attention to the sign changes, and combined the like terms. This process helps us correctly arrive at the missing addend. The key to mastering these types of problems is practicing, remembering the rules of signs, and consistently simplifying the terms. Remember, practice makes perfect. The more you work through these types of problems, the easier they'll become. Keep up the great work! This particular problem involved the fundamentals of algebraic thinking. The whole process of the subtraction of the polynomials and combining like terms and carefully evaluating the sign changes is a crucial skill to build in algebra. Keep practicing to enhance your skill in this area. You got this!