Which is true about the degree of the sum and difference of the polynomials and
?
Both the sum and difference have a degree of 6 .
Both the sum and difference have a degree of 7 .
The sum has a degree of 6 , but the difference has a degree of 7 .
The sum has a degree of 7 , but the difference has a degree of 6 .

Ask by Herrera Carroll. in the United States

Jan 24,2025

Question

Which is true about the degree of the sum and difference of the polynomials and
?
Both the sum and difference have a degree of 6 .
Both the sum and difference have a degree of 7 .
The sum has a degree of 6 , but the difference has a degree of 7 .
The sum has a degree of 7 , but the difference has a degree of 6 .

Ask by Herrera Carroll. in the United States

Jan 24,2025

UpStudy AI · Accepted Answer

To determine the degrees of the sum and difference of the given polynomials, let's analyze each step carefully.

**Given Polynomials:**
1. $$ P = 3x^5y - 2x^3y^4 - 7xy^3 $$
2. $$ Q = -8x^5y + 2x^3y^4 + xy^3 $$

**Sum $$ S = P + Q $$:**
$$
S = (3x^5y - 8x^5y) + (-2x^3y^4 + 2x^3y^4) + (-7xy^3 + xy^3)
$$
$$
S = -5x^5y + 0 + (-6xy^3) = -5x^5y - 6xy^3
$$
- **Degrees:**
  - $$ -5x^5y $$: Degree $$ 5 + 1 = 6 $$
  - $$ -6xy^3 $$: Degree $$ 1 + 3 = 4 $$
  
**Degree of Sum $$ S $$:** **6** (maximum degree among the terms)

---

**Difference $$ D = P - Q $$:**
$$
D = 3x^5y - (-8x^5y) + (-2x^3y^4) - 2x^3y^4 + (-7xy^3) - xy^3
$$
$$
D = 11x^5y - 4x^3y^4 - 8xy^3
$$
- **Degrees:**
  - $$ 11x^5y $$: Degree $$ 5 + 1 = 6 $$
  - $$ -4x^3y^4 $$: Degree $$ 3 + 4 = 7 $$
  - $$ -8xy^3 $$: Degree $$ 1 + 3 = 4 $$
  
**Degree of Difference $$ D $$:** **7** (maximum degree among the terms)

**Conclusion:**
- The **sum** has a degree of **6**.
- The **difference** has a degree of **7**.

**Answer:**
The sum has a degree of 6, but the difference has a degree of 7.
The sum has a degree of 6, but the difference has a degree of 7.

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