Let's classify each polynomial by combining the like terms first and then determining its degree.
1. Polynomial: [tex]\(2x - x^2 + 4x + x^2\)[/tex]
Step-by-step simplification:
[tex]\[
2x - x^2 + 4x + x^2
\][/tex]
Combine the like terms:
[tex]\[
(2x + 4x) + (-x^2 + x^2)
\][/tex]
Simplified:
[tex]\[
6x + 0 = 6x
\][/tex]
Degree: 1
2. Polynomial: [tex]\(x^3 - 5x^2 + 3 - 2\)[/tex]
Step-by-step simplification:
[tex]\[
x^3 - 5x^2 + 3 - 2
\][/tex]
Combine the like terms:
[tex]\[
x^3 - 5x^2 + (3 - 2)
\][/tex]
Simplified:
[tex]\[
x^3 - 5x^2 + 1
\][/tex]
Degree: 3
3. Polynomial: [tex]\(x^3 - x^3 + x^2 - x^2 + 3\)[/tex]
Step-by-step simplification:
[tex]\[
x^3 - x^3 + x^2 - x^2 + 3
\][/tex]
Combine the like terms:
[tex]\[
(x^3 - x^3) + (x^2 - x^2) + 3
\][/tex]
Simplified:
[tex]\[
0 + 0 + 3 = 3
\][/tex]
Degree: 0
4. Polynomial: [tex]\(7x - x^2 + x^3 + 4x^2\)[/tex]
Step-by-step simplification:
[tex]\[
7x - x^2 + x^3 + 4x^2
\][/tex]
Combine the like terms:
[tex]\[
x^3 + (-x^2 + 4x^2) + 7x
\][/tex]
Simplified:
[tex]\[
x^3 + 3x^2 + 7x
\][/tex]
Degree: 3
5. Polynomial: [tex]\(-6x - x^3 + 6x - 4x\)[/tex]
Step-by-step simplification:
[tex]\[
-6x - x^3 + 6x - 4x
\][/tex]
Combine the like terms:
[tex]\[
-x^3 + (-6x + 6x - 4x)
\][/tex]
Simplified:
[tex]\[
-x^3 - 4x
\][/tex]
Degree: 3
Now, let's summarize the classification of each polynomial with its degree:
1. [tex]\(6x\)[/tex], Degree: 1
2. [tex]\(x^3 - 5x^2 + 1\)[/tex], Degree: 3
3. [tex]\(3\)[/tex], Degree: 0
4. [tex]\(x^3 + 3x^2 + 7x\)[/tex], Degree: 3
5. [tex]\(-x^3 - 4x\)[/tex], Degree: 3
