Polynomial Subtraction: Fill In The Blanks!

October 11, 2025 by TextBrain Team 44 views

Hey math enthusiasts! Ever stumbled upon a polynomial subtraction problem that seemed to have a few missing pieces? Don't sweat it! It's like a puzzle, and we're here to help you fill in those blanks. Let's break down how to conquer these types of problems and make sure you're a polynomial subtraction whiz. We'll tackle the equation: $(9x^2 + oxed{ ext{ }}x + 13) - (2x^2 + 12x + oxed{ ext{ }}) = oxed{ ext{ }}x^2 + 5x + 2$ .

Understanding Polynomial Subtraction: The Basics

Alright, guys, before we dive into the nitty-gritty, let's get our heads around what polynomial subtraction actually is. Think of it like subtracting regular numbers, but with variables and exponents thrown into the mix. When we subtract polynomials, we're essentially subtracting the entire second polynomial from the entire first polynomial. The key here is to remember to subtract each term in the second polynomial. This often means changing the signs of the terms in the polynomial being subtracted. It’s super important to keep track of your signs—a little mistake here can throw off the whole answer! When you have a subtraction sign in front of a parenthesis, it changes the sign of everything inside the parenthesis. So, a positive term becomes negative, and a negative term becomes positive.

For example, if you see something like: $(a + b) - (c + d)$ , it becomes $a + b - c - d$ . See how the signs of c and d flipped? It's all about paying close attention to those negative signs. We combine like terms after applying the subtraction. Like terms are terms that have the same variable raised to the same power. You can only combine like terms, so $x^2$ terms can only be combined with other $x^2$ terms, $x$ terms with other $x$ terms, and constants with constants. We add or subtract the coefficients (the numbers in front of the variables) of the like terms, while keeping the variables and exponents the same. When you see a problem like the one we're trying to solve, it might look a little intimidating at first, but trust me, it’s totally manageable once you break it down step by step. Always remember that polynomial subtraction is a systematic process – you have to follow some basic rules, and with a little practice, you'll be acing these problems in no time. The main thing to remember is to be patient and work carefully.

Let's break down the example problem: $(9x^2 + oxed{ ext{ }}x + 13) - (2x^2 + 12x + oxed{ ext{ }}) = oxed{ ext{ }}x^2 + 5x + 2$ . The problem starts with the expression $9x^2$ and ends up with $x^2$ . That suggests that the $2x^2$ must have been subtracted, as we move from $9x^2$ to a lower number, in fact $7x^2$ . The term $12x$ in the second expression is what gets subtracted from some unknown term. From that operation, the result is $+5x$ . That suggests that $7x$ was the missing term. Finally, the constant term in the second expression, is what gives us a final constant term of $2$ . With all this information, we're ready to go! This should feel less intimidating, right?

Step-by-Step: Solving the Equation

Okay, let's get down to business and solve this equation step-by-step. We have: $(9x^2 + oxed{ ext{ }}x + 13) - (2x^2 + 12x + oxed{ ext{ }}) = oxed{ ext{ }}x^2 + 5x + 2$ . We'll work on filling in the blanks to complete the subtraction. Here is the step-by-step solution:

Focus on the $x^2$ terms: Look at the first terms in the polynomials. We have $9x^2$ in the first polynomial and $2x^2$ in the second. In the result, we need a $oxed{ ext{ }}x^2$ . So, let's subtract the coefficients: $9 - 2 = 7$ . Therefore, the coefficient for the $x^2$ term in the result is 7. So, the filled expression starts like this: $(9x^2 + oxed{ ext{ }}x + 13) - (2x^2 + 12x + oxed{ ext{ }}) = 7x^2 + 5x + 2$ .
Tackle the $x$ terms: Now, let's check out the $x$ terms. We have an unknown term $oxed{ ext{ }}x$ in the first polynomial, $12x$ in the second polynomial, and $5x$ in the result. We know that the second polynomial has to be subtracted from the first one, so we can work backward to find the missing value. We know that we have $-12x$ in the second term, and the result has $5x$ . So what number minus 12 gives us 5? The answer is 17. This is like solving the equation $? - 12 = 5$ . That gives us $17x$ as the missing term in the first polynomial. So, we update the equation: $(9x^2 + 17x + 13) - (2x^2 + 12x + oxed{ ext{ }}) = 7x^2 + 5x + 2$ .
Address the constant terms: Finally, let's address the constant terms. In the first polynomial, we have $13$ , in the second we have an unknown term, and in the result, we have $2$ . Subtracting the second polynomial means subtracting the constant term from 13, the result is 2. Therefore, we solve the equation $13 - ? = 2$ . That gives us 11. We plug it back to the formula: $(9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 7x^2 + 5x + 2$ .

So the completed equation looks like this: $(9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 7x^2 + 5x + 2$ . We've successfully filled in all the blanks!

Tips and Tricks for Success

Alright, you've got the basics down. Now, let's arm you with some extra tips and tricks to make sure you're a polynomial subtraction superstar. Here are a few pointers to keep in mind:

Double-Check Your Signs: Seriously, this is the golden rule. A missed negative sign can mess up the whole problem. Before you start combining terms, make sure you've correctly distributed the subtraction sign to every term inside the parentheses. A simple way to avoid this is to rewrite the problem by changing the sign of each term in the polynomial being subtracted and then adding the polynomials instead. For example, the subtraction problem above could be rewritten as $(9x^2 + 17x + 13) + (-2x^2 - 12x - 11)$ . Now the chance of making a mistake with signs is minimized.
Organize Your Work: When you're dealing with multiple terms and variables, things can get messy quickly. Write down your work in an organized way. Align like terms vertically to make it easier to combine them. You can also use different colors to highlight like terms or draw boxes around them. Keeping your work neat helps you avoid mistakes and makes it easier to go back and check your answer if you get stuck. The main thing here is to develop a system that works for you. This will not only help you with the problem at hand but also help you develop some great habits that will help you with more complex problems in the future.
Practice, Practice, Practice: The more you practice, the more comfortable you'll become with polynomial subtraction. Work through different examples and try to challenge yourself with more complex problems. Start with simple problems and gradually increase the difficulty. Don't get discouraged if you don't get it right away – everyone struggles when they're first learning something new. The key is to keep practicing and learning from your mistakes. There are tons of online resources, textbooks, and practice problems available to help you hone your skills. You can also try creating your own practice problems. Make sure that you understand the concepts behind the problems so that you can solve similar problems in the future.
Break It Down: Sometimes, problems can look overwhelming. Break them down into smaller, more manageable steps. Focus on one aspect of the problem at a time. Start with the $x^2$ terms, then move on to the $x$ terms, and finally, the constants. This approach can make the problem seem less daunting and help you avoid making careless errors.

Common Mistakes and How to Avoid Them

Even the best of us make mistakes sometimes. Let's look at some common errors to watch out for and how to avoid them.

Forgetting to Distribute the Negative Sign: This is, hands down, the most common mistake. Always remember to change the signs of every term in the second polynomial when you're subtracting. If you forget, your answer will be incorrect. The best way to avoid this is to rewrite the problem, changing the subtraction to addition and changing the sign of each term in the second polynomial. This makes it much easier to keep track of all the signs. Don't rush through this step – it’s worth taking a moment to ensure accuracy.
Combining Unlike Terms: You can only combine terms that are