Let's find the degree of each given polynomial step by step.
1. [tex]\((x^2 - y)^2\)[/tex]
First, expand the polynomial:
[tex]\[
(x^2 - y)^2 = (x^2 - y)(x^2 - y) = x^4 - 2x^2y + y^2
\][/tex]
Here, the term with the highest degree is [tex]\(x^4\)[/tex], so the degree is 4.
Degree: 4
2. [tex]\(8x^4 - 5x^7 + 4x^5\)[/tex]
Look at the degrees of each term:
[tex]\[
8x^4 \text{ has degree } 4\\
-5x^7 \text{ has degree } 7\\
4x^5 \text{ has degree } 5
\][/tex]
The highest degree term is [tex]\(-5x^7\)[/tex], so the degree is 7.
Degree: 7
3. [tex]\(\frac{x^2 y^3}{3} + 2x^3 + 8x^3\)[/tex]
Simplify the polynomial by combining like terms:
[tex]\[
\frac{x^2 y^3}{3} + (2x^3 + 8x^3) = \frac{x^2 y^3}{3} + 10x^3
\][/tex]
Now, look at the degrees of each term:
[tex]\[
\frac{x^2 y^3}{3} \text{ has degree } 2+3 = 5\\
10x^3 \text{ has degree } 3
\][/tex]
The highest degree term is [tex]\(\frac{x^2 y^3}{3}\)[/tex], so the degree is 5.
Degree: 5
4. [tex]\(x^4 + 2x^3 - 6x^4 - 17\)[/tex]
Simplify the polynomial by combining like terms:
[tex]\[
(x^4 - 6x^4) + 2x^3 - 17 = -5x^4 + 2x^3 - 17
\][/tex]
Now, look at the degrees of each term:
[tex]\[
-5x^4 \text{ has degree } 4\\
2x^3 \text{ has degree } 3\\
-17 \text{ (constant term) has degree } 0
\][/tex]
The highest degree term is [tex]\(-5x^4\)[/tex], so the degree is 4.
Degree: 4
5. [tex]\(-x^2 + 7x - 4x\)[/tex]
Simplify the polynomial by combining like terms:
[tex]\[
-x^2 + (7x - 4x) = -x^2 + 3x
\][/tex]
Now, look at the degrees of each term:
[tex]\[
-x^2 \text{ has degree } 2\\
3x \text{ has degree } 1
\][/tex]
The highest degree term is [tex]\(-x^2\)[/tex], so the degree is 2.
Degree: 2
6. [tex]\(x^7 + y^8 + x^7 - y^8\)[/tex]
Simplify the polynomial by combining like terms:
[tex]\[
(x^7 + x^7) + (y^8 - y^8) = 2x^7 + 0 = 2x^7
\][/tex]
Now, look at the degrees of each term:
[tex]\[
2x^7 \text{ has degree } 7
\][/tex]
The highest degree term is [tex]\(2x^7\)[/tex], so the degree is 7.
Degree: 7
In summary:
[tex]\[
\begin{array}{l}
\text{Degree } 4 \\
\text{Degree } 7 \\
\text{Degree } 5 \\
\text{Degree } 4 \\
\text{Degree } 2 \\
\text{Degree } 7 \\
\end{array}
\][/tex]
