giannisdagoat
Answered

Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

Which is true about the degree of the sum and difference of the polynomials [tex]\(3x^5y - 2x^3y^4 - 7xy^3\)[/tex] and [tex]\(-8x^5y + 2x^3y^4 + xy^3\)[/tex]?

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 has a degree of 6, but the difference has a degree of 7.
D. The sum has a degree of 7, but the difference has a degree of 6.

Sagot :

GinnyAnswer
To determine the degree of the sum and difference of the given polynomials, we start by examining the polynomials themselves:

First polynomial:
[tex]\[ 3x^5 y - 2x^3 y^4 - 7xy^3 \][/tex]

Second polynomial:
[tex]\[ -8x^5 y + 2x^3 y^4 + xy^3 \][/tex]

### Step 1: Sum of the Polynomials
We add the corresponding terms of the two polynomials:

[tex]\[ (3x^5 y - 2x^3 y^4 - 7xy^3) + (-8x^5 y + 2x^3 y^4 + xy^3) \][/tex]

Combine like terms:
[tex]\[ (3x^5y - 8x^5y) + (-2x^3y^4 + 2x^3y^4) + (-7xy^3 + xy^3) \][/tex]
[tex]\[ -5x^5y + 0x^3y^4 - 6xy^3 \][/tex]

The simplified sum polynomial is:
[tex]\[ -5x^5 y - 6xy^3 \][/tex]

The degree of a polynomial is determined by the term with the highest sum of exponents (the degree of each term is found by adding the exponents of its variables).

- [tex]\( -5x^5 y \)[/tex] has a degree of [tex]\(5 + 1 = 6\)[/tex]
- [tex]\( -6xy^3 \)[/tex] has a degree of [tex]\(1 + 3 = 4\)[/tex]

The highest degree is 6, so the sum of the polynomials has a degree of 6.

### Step 2: Difference of the Polynomials
We subtract the second polynomial from the first:

[tex]\[ (3x^5 y - 2x^3 y^4 - 7xy^3) - (-8x^5 y + 2x^3 y^4 + xy^3) \][/tex]

Distribute the negative sign:
[tex]\[ 3x^5 y - 2x^3 y^4 - 7xy^3 + 8x^5 y - 2x^3 y^4 - xy^3 \][/tex]

Combine like terms:
[tex]\[ (3x^5 y + 8x^5 y) + (-2x^3 y^4 - 2x^3 y^4) + (-7xy^3 - xy^3) \][/tex]
[tex]\[ 11x^5 y - 4x^3 y^4 - 8xy^3 \][/tex]

The simplified difference polynomial is:
[tex]\[ 11x^5 y - 4x^3 y^4 - 8xy^3 \][/tex]

Now, check the degree of each term:

- [tex]\( 11x^5 y \)[/tex] has a degree of [tex]\(5 + 1 = 6\)[/tex]
- [tex]\( -4x^3 y^4 \)[/tex] has a degree of [tex]\(3 + 4 = 7\)[/tex]
- [tex]\( -8xy^3 \)[/tex] has a degree of [tex]\(1 + 3 = 4\)[/tex]

The highest degree is 7, so the difference of the polynomials has a degree of 7.

### Conclusion
From the steps above, we have determined:
- The degree of the sum of the polynomials is 6.
- The degree of the difference of the polynomials is 7.

Thus, the correct statement is:
"The sum has a degree of 6, but the difference has a degree of 7."
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.