Polynomial Degrees Exploring Sums And Differences

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of polynomials, specifically exploring how their degrees behave when we add or subtract them. We'll be tackling a specific problem, but more importantly, we'll be equipping you with the knowledge to conquer similar challenges. So, buckle up and let's get started!

The Polynomial Puzzle: Degrees of Sums and Differences

Our main focus is to find which statement accurately describes the degree of both the sum and the difference of the following polynomials:

Polynomial 1: $3x^5y - 2x^3y^4 - 7xy^3$ Polynomial 2: $-8x^5y + 2x^3y^4 + xy^3$

We have three options to consider:

A. Both the sum and difference have a degree of 6. B. Both the sum and difference have a degree of 7. C. The sum...

To solve this, we'll need to understand a few key concepts about polynomials and their degrees. Don't worry, we'll break it down step by step!

Decoding Polynomial Degrees: A Quick Refresher

Before we jump into the problem, let's quickly review what the degree of a polynomial actually means. Understanding polynomial degrees is crucial for determining how polynomials behave and interact with each other, especially when performing operations like addition and subtraction. The degree of a polynomial term is the sum of the exponents of its variables. For example:

• The term $3x^2$ has a degree of 2 (because the exponent of x is 2).
• The term $5xy^3$ has a degree of 4 (because the exponents are 1 for x and 3 for y, and 1 + 3 = 4).
• The term $-7$ (a constant) has a degree of 0 (think of it as $-7x^0$).

Now, the degree of the entire polynomial is the highest degree among all its terms. Let's look at our example polynomials again:

Polynomial 1: $3x^5y - 2x^3y^4 - 7xy^3$

• Term $3x^5y$ has a degree of 5 + 1 = 6
• Term $-2x^3y^4$ has a degree of 3 + 4 = 7
• Term $-7xy^3$ has a degree of 1 + 3 = 4

The highest degree among these is 7, so the degree of Polynomial 1 is 7.

Polynomial 2: $-8x^5y + 2x^3y^4 + xy^3$

• Term $-8x^5y$ has a degree of 5 + 1 = 6
• Term $2x^3y^4$ has a degree of 3 + 4 = 7
• Term $xy^3$ has a degree of 1 + 3 = 4

Similarly, the degree of Polynomial 2 is also 7.

Summing It Up: Adding Polynomials

Now that we know how to find the degree of a polynomial, let's tackle the sum of our two polynomials. To add polynomials, we simply combine like terms (terms with the same variables raised to the same powers). It’s like grouping similar objects together – you can add apples to apples, but you can't directly add apples to oranges. In the polynomial world, $x^2$ and $x^2$ are like terms, while $x^2$ and $x^3$ are not.

So, let's add Polynomial 1 and Polynomial 2:

$(3x^5y - 2x^3y^4 - 7xy^3) + (-8x^5y + 2x^3y^4 + xy^3)$

Combine the like terms:

$(3x^5y - 8x^5y) + (-2x^3y^4 + 2x^3y^4) + (-7xy^3 + xy^3)$

This simplifies to:

$-5x^5y + 0x^3y^4 - 6xy^3$

Which is just:

$-5x^5y - 6xy^3$

Now, let's find the degree of this sum:

• Term $-5x^5y$ has a degree of 5 + 1 = 6
• Term $-6xy^3$ has a degree of 1 + 3 = 4

The highest degree is 6, so the degree of the sum is 6.

The Difference Maker: Subtracting Polynomials

Next, we need to find the difference between the two polynomials. Subtracting polynomials is similar to adding, but we need to be careful with the signs. We're essentially distributing a negative sign across the second polynomial before combining like terms. Think of it as subtracting the entire group, not just the first term.

So, let's subtract Polynomial 2 from Polynomial 1:

$(3x^5y - 2x^3y^4 - 7xy^3) - (-8x^5y + 2x^3y^4 + xy^3)$

Distribute the negative sign:

$3x^5y - 2x^3y^4 - 7xy^3 + 8x^5y - 2x^3y^4 - xy^3$

Now, combine the like terms:

$(3x^5y + 8x^5y) + (-2x^3y^4 - 2x^3y^4) + (-7xy^3 - xy^3)$

This simplifies to:

$11x^5y - 4x^3y^4 - 8xy^3$

Let's find the degree of this difference:

• Term $11x^5y$ has a degree of 5 + 1 = 6
• Term $-4x^3y^4$ has a degree of 3 + 4 = 7
• Term $-8xy^3$ has a degree of 1 + 3 = 4

The highest degree is 7, so the degree of the difference is 7.

The Grand Reveal: Our Solution

Alright, we've done the heavy lifting! We found that:

• The degree of the sum of the polynomials is 6.
• The degree of the difference of the polynomials is 7.

Looking back at our options:

A. Both the sum and difference have a degree of 6. B. Both the sum and difference have a degree of 7. C. The sum...

We can see that option A is incorrect because the difference has a degree of 7. Option B is also incorrect because the sum has a degree of 6. To find out the correct options, we need the complete options.

Key Takeaways and Pro Tips

• The degree of a polynomial is the highest degree of its terms.
• To add or subtract polynomials, combine like terms.
• When subtracting, remember to distribute the negative sign.
• The degree of the sum or difference isn't always the same as the degree of the original polynomials, especially when terms cancel out.

Understanding these concepts will make you a polynomial pro in no time! Keep practicing, and you'll be solving these puzzles with ease.

Practice Makes Perfect: Test Your Knowledge

Now that you've mastered the art of finding the degree of sums and differences of polynomials, why not put your skills to the test? Try solving similar problems on your own. You can find plenty of practice questions online or in your textbook. Remember, the more you practice, the more confident you'll become!

And hey, if you ever get stuck, don't hesitate to review the steps we covered today. We broke down the process into manageable chunks, so you can always refer back to the explanations and examples. Keep up the great work, and happy problem-solving!