Simplifying Polynomials: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of polynomials, specifically tackling the question: "Which statement is true about the polynomial $-3 x^4 y^3+8 x y^5-3+18 x^3 y^4-3 x y^5$ after it has been fully simplified?" Don't worry, it might look a bit intimidating at first, but trust me, simplifying polynomials is a pretty straightforward process once you get the hang of it. Let's break it down, step by step, and figure out the correct answer from the multiple-choice options. We'll explore how to combine like terms, determine the number of terms, and identify the degree of a polynomial. By the end of this guide, you'll be a pro at simplifying polynomials, ready to tackle any problem that comes your way. So, buckle up, grab your pens and paper, and let's get started!

Understanding the Basics: What are Polynomials?

Before we jump into the nitty-gritty of simplifying, let's make sure we're all on the same page about what a polynomial actually is. Polynomials are algebraic expressions made up of terms. Each term consists of a constant (a number), one or more variables (like x and y), and exponents (the little numbers above the variables). Think of them as building blocks, with different combinations creating different polynomials. For instance, in our example, $-3 x^4 y^3+8 x y^5-3+18 x^3 y^4-3 x y^5$, each part separated by a plus or minus sign is a term. So, we have $-3x^4y^3$, $8xy^5$, $-3$, $18x^3y^4$, and $-3xy^5$. Polynomials can have one term (monomial), two terms (binomial), three terms (trinomial), or many terms (polynomials with more than three terms are just called polynomials!). The key is that the exponents on the variables must be non-negative integers (0, 1, 2, 3, and so on). You won't find things like fractional or negative exponents in a polynomial. Got it? Great, now we can move on to the fun part!

Step 1: Combining Like Terms - The Key to Simplification

Alright guys, the first and most crucial step in simplifying any polynomial is to combine like terms. Like terms are terms that have the exact same variables raised to the exact same powers. Think of it like this: you can only add or subtract things that are the same. You wouldn't try to add apples and oranges, right? Similarly, you can only combine terms with the same variables and exponents. In our example polynomial, $-3 x^4 y^3+8 x y^5-3+18 x^3 y^4-3 x y^5$, let's look for like terms. We have $8xy^5$ and $-3xy^5$, which are like terms because they both have 'x' to the power of 1 and 'y' to the power of 5. These are the only like terms we have. Now, let's combine them: $8xy^5 - 3xy^5 = 5xy^5$. Remember, when combining like terms, you only add or subtract the coefficients (the numbers in front of the variables). The variables and their exponents stay exactly the same. So after we combine those like terms, our polynomial becomes: $-3x^4y^3 + 5xy^5 - 3 + 18x^3y^4$. Notice that -3 is a constant, which is a like term with other constants, but there are no other constants to combine it with. Now it is simplified and ready to identify the number of terms and the degree.

Step 2: Identifying the Number of Terms - Counting the Pieces

Once you've combined all the like terms, the next step is to count the number of terms in your simplified polynomial. Remember, terms are separated by plus or minus signs. In our simplified polynomial, $-3x^4y^3 + 5xy^5 - 3 + 18x^3y^4$, let's count: We have $-3x^4y^3$, $5xy^5$, $-3$, and $18x^3y^4$. That’s four different chunks, right? So, after simplification, this polynomial has four terms. This is a crucial piece of information for answering our original question, as the answer choices refer to the number of terms.

Step 3: Determining the Degree of a Polynomial - Finding the Highest Power

Last but not least, we need to figure out the degree of the polynomial. The degree is the highest power of the variable (or sum of powers if there are multiple variables in a term) in the polynomial. To find the degree, look at each term individually and add up the exponents of the variables in that term. Then, the highest sum you find is the degree of the polynomial. Let's go through it for each term in our simplified polynomial, $-3x^4y^3 + 5xy^5 - 3 + 18x^3y^4$.

• For the term $-3x^4y^3$, the exponents are 4 and 3. Adding them together, we get 4 + 3 = 7.
• For the term $5xy^5$, the exponents are 1 and 5. Adding them together, we get 1 + 5 = 6.
• The term $-3$ is a constant, and constants have a degree of 0 (because there are no variables).
• For the term $18x^3y^4$, the exponents are 3 and 4. Adding them together, we get 3 + 4 = 7.

Now, compare the results: 7, 6, 0, and 7. The highest of these is 7. So, the degree of the simplified polynomial is 7. This is another crucial piece of information because the answer choices mention the degree of the polynomial. This step helps us choose the correct answer by comparing our calculated degree with the options provided.

Step 4: Putting It All Together - Choosing the Correct Answer

Okay, we've done all the hard work! Now, let's go back to our multiple-choice options and see which one matches our findings:

• A. It has 3 terms and a degree of 5.
• B. It has 3 terms and a degree of 7.
• C. It has 4 terms and a degree of 5.
• D. It has 4 terms and a degree of 7.

We found that the simplified polynomial has 4 terms and a degree of 7. Looking at the options, we can see that option D is the correct answer. Congratulations, you did it!

Conclusion: Mastering Polynomials

Great job guys! You've successfully simplified a polynomial and found the correct answer. Remember that simplifying polynomials is all about combining like terms, counting the number of terms, and finding the degree. Keep practicing, and you'll become a pro in no time. If you have any questions, feel free to ask. Keep up the great work, and happy simplifying!