To classify each polynomial as constant, linear, quadratic, or cubic, we need to combine the like terms first. Let’s analyze each polynomial step by step:
1. Polynomial: [tex]\( x^3 - 2x + x^3 \)[/tex]
- Combine the like terms:
[tex]\[
x^3 + x^3 - 2x = 2x^3 - 2x
\][/tex]
- Degree of the polynomial:
- The highest power of [tex]\( x \)[/tex] is 3.
- Classification:
[tex]\[
\text{Cubic}
\][/tex]
2. Polynomial: [tex]\( 4x^2 - 6x - 8x^2 \)[/tex]
- Combine the like terms:
[tex]\[
4x^2 - 8x^2 - 6x = -4x^2 - 6x
\][/tex]
- Degree of the polynomial:
- The highest power of [tex]\( x \)[/tex] is 2.
- Classification:
[tex]\[
\text{Quadratic}
\][/tex]
3. Polynomial: [tex]\( 6x - 6 + 6x \)[/tex]
- Combine the like terms:
[tex]\[
6x + 6x - 6 = 12x - 6
\][/tex]
- Degree of the polynomial:
- The highest power of [tex]\( x \)[/tex] is 1.
- Classification:
[tex]\[
\text{Linear}
\][/tex]
4. Polynomial: [tex]\( 5 + 4x^2 - 4x^2 + 5 \)[/tex]
- Combine the like terms:
[tex]\[
5 + 5 + 4x^2 - 4x^2 = 10
\][/tex]
- Degree of the polynomial:
- There is no [tex]\( x \)[/tex] term, so the degree is 0.
- Classification:
[tex]\[
\text{Constant}
\][/tex]
Therefore, the classifications are:
1. [tex]\( 2x^3 - 2x \)[/tex] → Cubic
2. [tex]\( -4x^2 - 6x \)[/tex] → Quadratic
3. [tex]\( 12x - 6 \)[/tex] → Linear
4. [tex]\( 10 \)[/tex] → Constant
