Simplifying Polynomials: Finding The True Statement

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Simplifying Polynomials: Finding the True Statement

Hey everyone! Today, we're diving into the world of polynomials. We'll be tackling a problem that involves simplifying a polynomial expression and figuring out which statement about it is true. This stuff is super important, so let's get started. Get ready to flex those math muscles!

Understanding the Basics: Polynomials and Their Parts

Alright, before we jump into the problem, let's brush up on what polynomials are all about. Think of a polynomial as a mathematical expression made up of terms. Each term can be a number (like 2, 5, or -3), a variable (like x or y), or a combination of both, like 2x or -5xy². These terms are connected by addition or subtraction.

Now, let's talk about the degree of a term. The degree is the sum of the exponents of the variables in that term. For example, in the term 3x²y², the degree is 2 + 2 = 4 (because x is raised to the power of 2, and y is also raised to the power of 2). In the term -5xy², the degree is 1 + 2 = 3 (because x is raised to the power of 1, and y is raised to the power of 2). For a term that's just a constant number (like 2), the degree is 0.

The degree of a polynomial is the highest degree among all its terms. For example, the polynomial 4x³ + 2x² - x + 5 has a degree of 3 (because the term 4x³ has the highest degree). Lastly, the number of terms is simply how many parts are added or subtracted together. In our example 4x³ + 2x² - x + 5 there are 4 terms.

So, to recap: Polynomials are built from terms, each term has a degree, and the degree of the polynomial is the highest degree of its terms. We also can easily find the number of terms by counting each part added or subtracted from each other. Got it? Awesome! Let's simplify and figure this problem out.

Breaking Down the Question

So, the question gives us a polynomial: 3x²y² - 5xy² - 3x²y² + 2x*². We need to simplify this expression first. Then, we need to figure out which of the answer choices is true about the simplified form. The options are about the number of terms and the degree of the polynomial. So, let's simplify and see what we get.

Simplifying the Polynomial Expression: Step-by-Step

Alright, let's get our hands dirty and simplify the polynomial: 3x²y² - 5xy² - 3x²y² + 2x². The key to simplifying polynomials is to combine like terms. Like terms are terms that have the same variables raised to the same powers. For example, 3x²y² and -3x²y² are like terms. Similarly, if there were any other term like 2x*², it would be considered like terms.

  1. Identify Like Terms:

    • We can see that 3x²y² and -3x²y² are like terms. Also, there is a term of 2x².
    • The term -5xy² does not have any like terms.

  2. Combine Like Terms:

    • Combine 3x²y² and -3x²y²: 3x²y² - 3x²y² = 0.
    • The term -5xy² remains as is.
    • The term 2x² remains as is.

  3. Write the Simplified Expression:

    • After combining like terms, our expression simplifies to: 0 - 5xy² + 2x² or simply -5xy² + 2x².

Analyzing the Simplified Polynomial

After simplifying, we have the polynomial -5xy² + 2x². Now, let's analyze it to determine the number of terms and the degree of the polynomial. This is where we extract the values to answer the problem.

  1. Number of Terms:

    • The simplified polynomial has two terms: -5xy² and 2x². Terms are separated by addition or subtraction signs. We have two separate parts here, so there are two terms.

  2. Degree of the Polynomial:

    • Let's find the degree of each term.
      • The degree of -5xy² is 1 + 2 = 3 (because x is raised to the power of 1, and y is raised to the power of 2).
      • The degree of 2x² is 2 (because x is raised to the power of 2).
    • The degree of the polynomial is the highest degree of its terms, which is 3.

Choosing the Correct Answer: Let's Get It Right!

Now that we've simplified the polynomial and analyzed its components, we can easily pick the correct answer. The simplified polynomial, -5xy² + 2x², has:

  • 2 terms
  • A degree of 3.

Therefore, the correct answer must state these facts. Let's look back at the answer options to find the one that matches our findings.

Reviewing the Answer Choices

Let's go through the answer choices one by one:

A. It has 2 terms and a degree of 2.

B. It has 2 terms and a degree of 3.

C. It has 4 terms and a degree of 2.

D. It has 4 terms and a degree of 3.

Answer choice B is the correct one. It states that the polynomial has 2 terms and a degree of 3, which is what we found after simplifying the expression. It is important to go back and check your work to ensure you do not make any errors in your process.

Final Thoughts: Mastering Polynomials

Alright, guys, there you have it! We successfully simplified a polynomial, figured out its degree, and determined the number of terms. Polynomial problems like this are fundamental, so it's important to grasp the concepts and be confident in solving them. Remember to focus on combining like terms and understanding the degree of each term. Keep practicing, and you'll become a polynomial pro in no time! Remember to always stay curious, keep learning, and never be afraid to ask for help. See you in the next lesson!

Tips for Success

  • Practice, practice, practice: The more you work with polynomials, the more comfortable you'll become.
  • Pay attention to signs: Don't let those negative signs trip you up!
  • Double-check your work: Always go back and make sure you haven't missed any terms or made any calculation errors.
  • Ask for help: If you're struggling, don't hesitate to ask your teacher, classmates, or a tutor for help.

Thanks for tuning in! Keep up the great work, and remember, math is a journey, not a destination. Keep exploring and enjoying the world of numbers and equations! Feel free to ask if you have any questions!