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.
