Classify each polynomial as constant, linear, quadratic, or cubic. Combine like terms first.

1. [tex] x^3 - 2x + x^3 [/tex]
[tex] \square [/tex]

2. [tex] x^3 + 3x^3 + 2 [/tex]
[tex] \square [/tex]

3. [tex] 4x^2 - 6x - 8x^2 [/tex]
[tex] \square [/tex]

4. [tex] 2x^3 + 5x + 3 [/tex]
[tex] \square [/tex]

5. [tex] 6x - 6 + 6x [/tex]
[tex] \square [/tex]

6. [tex] 4x - 5x + x [/tex]
[tex] \square [/tex]

7. [tex] 5 + 4x^2 - 4x^2 + 5 [/tex]
[tex] \square [/tex]

8. [tex] 6x^2 + 5 - 2x [/tex]
[tex] \square [/tex]



Answer :

Let's classify each polynomial by first combining like terms and then identifying their degrees. The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] in the expression, which helps us classify the polynomial as either constant, linear, quadratic, or cubic.

1. Polynomial: [tex]\( x^3 - 2x + x^3 \)[/tex]
- Combine like terms:
[tex]\[ x^3 + x^3 - 2x = 2x^3 - 2x \][/tex]
- Classification: The highest power of [tex]\( x \)[/tex] here is 3, hence it is a cubic polynomial.

2. Polynomial: [tex]\( 4x^2 - 6x - 8x^2 \)[/tex]
- Combine like terms:
[tex]\[ 4x^2 - 8x^2 - 6x = -4x^2 - 6x \][/tex]
- Classification: The highest power of [tex]\( x \)[/tex] here is 2, hence it is a quadratic polynomial.

3. Polynomial: [tex]\( 6x - 6 + 6x \)[/tex]
- Combine like terms:
[tex]\[ 6x + 6x - 6 = 12x - 6 \][/tex]
- Classification: The highest power of [tex]\( x \)[/tex] here is 1, hence it is a linear polynomial.

4. Polynomial: [tex]\( 5 + 4x^2 - 4x^2 + 5 \)[/tex]
- Combine like terms:
[tex]\[ 4x^2 - 4x^2 + 5 + 5 = 0 + 10 = 10 \][/tex]
- Classification: This polynomial simplifies to a constant value of 10, hence it is a constant polynomial.

Here is the final classification of each polynomial:

1. [tex]\( x^3 - 2x + x^3 \)[/tex] is cubic.
2. [tex]\( 4x^2 - 6x - 8x^2 \)[/tex] is quadratic.
3. [tex]\( 6x - 6 + 6x \)[/tex] is linear.
4. [tex]\( 5 + 4x^2 - 4x^2 + 5 \)[/tex] is constant.