To find the simplified difference of the given polynomials [tex]\( 6x^6 - x^3 y^4 - 5xy^5 \)[/tex] and [tex]\( 4x^5 y + 2x^3 y^4 + 5xy^5 \)[/tex], we will follow these steps:
1. Write the polynomials:
[tex]\[
P_1 = 6x^6 - x^3 y^4 - 5xy^5
\][/tex]
[tex]\[
P_2 = 4x^5 y + 2x^3 y^4 + 5xy^5
\][/tex]
2. Calculate the difference between the polynomials:
[tex]\[
P = P_1 - P_2 = (6x^6 - x^3 y^4 - 5xy^5) - (4x^5 y + 2x^3 y^4 + 5xy^5)
\][/tex]
3. Distribute and combine like terms:
[tex]\[
P = 6x^6 - x^3 y^4 - 5xy^5 - 4x^5 y - 2x^3 y^4 - 5xy^5
\][/tex]
Combine the terms with the same degree and exponents:
- [tex]\(6x^6\)[/tex]
- Combine the [tex]\(x^3 y^4\)[/tex] terms: [tex]\(-x^3 y^4 - 2x^3 y^4 = -3x^3 y^4\)[/tex]
- Combine the [tex]\(xy^5\)[/tex] terms: [tex]\(-5xy^5 - 5xy^5 = -10xy^5\)[/tex]
- [tex]\(-4x^5 y\)[/tex]
Therefore:
[tex]\[
P = 6x^6 - 4x^5 y - 3x^3 y^4 - 10xy^5
\][/tex]
4. Count the number of terms in the simplified polynomial:
The polynomial [tex]\( P \)[/tex] has four terms:
- [tex]\(6x^6\)[/tex]
- [tex]\(-4x^5 y\)[/tex]
- [tex]\(-3x^3 y^4\)[/tex]
- [tex]\(-10xy^5\)[/tex]
5. Determine the degree of the simplified polynomial:
The degree of a polynomial is the highest degree of its terms. Evaluate the degrees of each term:
- The degree of [tex]\(6x^6\)[/tex] is 6.
- The degree of [tex]\(-4x^5 y\)[/tex] is [tex]\(5 + 1 = 6\)[/tex].
- The degree of [tex]\(-3x^3 y^4\)[/tex] is [tex]\(3 + 4 = 7\)[/tex].
- The degree of [tex]\(-10xy^5\)[/tex] is [tex]\(1 + 5 = 6\)[/tex].
The highest degree among them is 7.
Therefore, the completely simplified difference of the polynomials has 4 terms and a degree of 7. The correct answer is:
- The difference has 4 terms and a degree of 7.
