Let's group the given equations based on their common properties. We observe that some equations are polynomial equations, which only involve variables raised to integer powers and potentially constants. Others are linear equations, where variables are raised only to the power of one (with potential constants).
### Group 1: Polynomial Equations
These equations involve variables raised to various powers (e.g., squares, cubes):
1. [tex]\( n^3 - 3n + 10 = 0 \)[/tex]
2. [tex]\( 2x^3 + 2x + 1 = 0 \)[/tex]
3. [tex]\( a^2 = 225 \)[/tex]
4. [tex]\( 25b^2 - 16 = 0 \)[/tex]
5. [tex]\( F^2 - 3F + 2 = 0 \)[/tex]
6. [tex]\( \frac{1}{3}m^2 + 2m = 4 \)[/tex]
### Group 2: Linear Equations
These equations involve variables raised to only the first power:
1. [tex]\( k = \frac{4}{3} \)[/tex]
2. [tex]\( 3r + 2c = -6 \)[/tex]
3. [tex]\( 2y - 2 = 9 \)[/tex]
4. [tex]\( 10c - 5 = 8 \)[/tex]
5. [tex]\( 5w + 5 = 0 \)[/tex]
6. [tex]\( 8 - 3k = 12 \)[/tex]
7. [tex]\( d = 3c - 7 \)[/tex]
### Re-arranged groups:
#### Polynomial Equations
1. [tex]\( n^3 - 3n + 10 = 0 \)[/tex]
2. [tex]\( 2x^3 + 2x + 1 = 0 \)[/tex]
3. [tex]\( a^2 = 225 \)[/tex]
4. [tex]\( 25b^2 - 16 = 0 \)[/tex]
5. [tex]\( F^2 - 3F + 2 = 0 \)[/tex]
6. [tex]\( \frac{1}{3}m^2 + 2m = 4 \)[/tex]
#### Linear Equations
1. [tex]\( k = \frac{4}{3} \)[/tex]
2. [tex]\( 3r + 2c = -6 \)[/tex]
3. [tex]\( 2y - 2 = 9 \)[/tex]
4. [tex]\( 10c - 5 = 8 \)[/tex]
5. [tex]\( 5w + 5 = 0 \)[/tex]
6. [tex]\( d = 3c - 7 \)[/tex]
7. [tex]\( 8 - 3k = 12 \)[/tex]
In summary, polynomial equations include variables with powers of 2 or higher, while linear equations include variables raised only to the power of one and constants.
