Alright, let's classify and simplify each of the given polynomials by combining like terms. Follow the steps below to obtain the simplified forms.
### Polynomial 1: [tex]\(2x - x^2 + 4x + x^2\)[/tex]
1. Identify and combine like terms:
[tex]\[
2x + 4x + (-x^2) + x^2
\][/tex]
2. Combine the coefficients of [tex]\(x\)[/tex]:
[tex]\[
(2 + 4)x = 6x
\][/tex]
3. Combine the coefficients of [tex]\(x^2\)[/tex]:
[tex]\[
(-1 + 1)x^2 = 0x^2 = 0
\][/tex]
4. The simplified polynomial is:
[tex]\[
6x
\][/tex]
### Polynomial 2: [tex]\(x^3 - 5x^2 + 3 - 2\)[/tex]
1. Identify and combine like terms:
[tex]\[
x^3 + (-5x^2) + (3 - 2)
\][/tex]
2. Simplify the constant terms:
[tex]\[
3 - 2 = 1
\][/tex]
3. The simplified polynomial is:
[tex]\[
x^3 - 5x^2 + 1
\][/tex]
### Polynomial 3: [tex]\(x^3 - x^3 + x^2 - x^2 + 3\)[/tex]
1. Identify and combine like terms:
[tex]\[
(x^3 - x^3) + (x^2 - x^2) + 3
\][/tex]
2. Combine the coefficients of [tex]\(x^3\)[/tex]:
[tex]\[
x^3 - x^3 = 0
\][/tex]
3. Combine the coefficients of [tex]\(x^2\)[/tex]:
[tex]\[
x^2 - x^2 = 0
\][/tex]
4. The simplified polynomial is:
[tex]\[
0 + 0 + 3 = 3
\][/tex]
### Polynomial 4: [tex]\(7x - x^2 + x^3 + 4x^2\)[/tex]
1. Identify and combine like terms:
[tex]\[
7x + (-x^2 + 4x^2) + x^3
\][/tex]
2. Combine the coefficients of [tex]\(x^2\)[/tex]:
[tex]\[
-x^2 + 4x^2 = 3x^2
\][/tex]
3. The simplified polynomial is:
[tex]\[
x^3 + 3x^2 + 7x
\][/tex]
### Polynomial 5: [tex]\(-6x - x^3 + 6x - 4x\)[/tex]
1. Identify and combine like terms:
[tex]\[
-6x + 6x - 4x - x^3
\][/tex]
2. Combine the coefficients of [tex]\(x\)[/tex]:
[tex]\[
(-6 + 6 - 4)x = -4x
\][/tex]
3. The simplified polynomial is:
[tex]\[
-x^3 - 4x
\][/tex]
So, after combining like terms, the classified polynomials are:
[tex]\[
\begin{array}{c}
6x \\
x^3 - 5x^2 + 1 \\
3 \\
x^3 + 3x^2 + 7x \\
-x^3 - 4x \\
\end{array}
\][/tex]
