Sure, let's classify each polynomial step-by-step by first combining like terms, and then determining its degree or the number of terms.
### Part 1: Classification by Degree
1. [tex]\( x^3 - 2x + x^3 \)[/tex]
Combine like terms:
[tex]\[
x^3 + x^3 - 2x = 2x^3 - 2x
\][/tex]
The highest power is [tex]\( x^3 \)[/tex], so this is a cubic polynomial.
2. [tex]\( 4x^2 - 6x - 8x^2 \)[/tex]
Combine like terms:
[tex]\[
4x^2 - 8x^2 - 6x = -4x^2 - 6x
\][/tex]
The highest power is [tex]\( x^2 \)[/tex], so this is a quadratic polynomial.
3. [tex]\( 6x - 6 + 6x \)[/tex]
Combine like terms:
[tex]\[
6x + 6x - 6 = 12x - 6
\][/tex]
The highest power is [tex]\( x \)[/tex], so this is a linear polynomial.
4. [tex]\( 5 + 4x^2 - 4x^2 + 5 \)[/tex]
Combine like terms:
[tex]\[
5 + 5 + 4x^2 - 4x^2 = 10
\][/tex]
There are no [tex]\( x \)[/tex] terms present; it's just [tex]\( 10 \)[/tex], which is a constant. So this is a constant polynomial.
### Part 2: Classification by Number of Terms
1. [tex]\( x^3 + 3x^3 + 2x \)[/tex]
Combine like terms:
[tex]\[
x^3 + 3x^3 + 2x = 4x^3 + 2x
\][/tex]
There are two terms: [tex]\( 4x^3 \)[/tex] and [tex]\( 2x \)[/tex]. So this is a binomial.
2. [tex]\( 2x^3 + 5x + 3x^4 - x \)[/tex]
Combine like terms:
[tex]\[
2x^3 + 3x^4 + 4x - x = 3x^4 + 2x^3 + 4x
\][/tex]
There are three distinct terms: [tex]\( 3x^4 \)[/tex], [tex]\( 2x^3 \)[/tex], and [tex]\( 4x \)[/tex]. So this is a trinomial.
3. [tex]\( 4x - 5x + x - 2 \)[/tex]
Combine like terms:
[tex]\[
4x - 5x + x - 2 = 0 - 2 = -2
\][/tex]
After simplifying the polynomial, we are left with a constant, -2. Despite there originally being four terms, after simplification, it is a single term. So this is a monomial.
4. [tex]\( 6x^2 + 5 - 2x^2 - 9 \)[/tex]
Combine like terms:
[tex]\[
6x^2 - 2x^2 + 5 - 9 = 4x^2 - 4
\][/tex]
There are two terms: [tex]\( 4x^2 \)[/tex] and [tex]\(-4\)[/tex]. So this is a binomial.
### Summary:
1. [tex]\( x^3 - 2x + x^3 \)[/tex] - Cubic
2. [tex]\( 4x^2 - 6x - 8x^2 \)[/tex] - Quadratic
3. [tex]\( 6x - 6 + 6x \)[/tex] - Linear
4. [tex]\( 5 + 4x^2 - 4x^2 + 5 \)[/tex] - Constant
5. [tex]\( x^3 + 3x^3 + 2x \)[/tex] - Binomial
6. [tex]\( 2x^3 + 5x + 3x^4 - x \)[/tex] - Trinomial
7. [tex]\( 4x - 5x + x - 2 \)[/tex] - Monomial
8. [tex]\( 6x^2 + 5 - 2x^2 - 9 \)[/tex] - Binomial
