Certainly! Let's classify each polynomial step-by-step by combining like terms first.
### Polynomial 1: [tex]\( x^3 + 3x^3 + 2x \)[/tex]
1. Combine like terms:
[tex]\[
x^3 + 3x^3 = 4x^3 \quad \text{and} \quad 2x
\][/tex]
So, the simplified form is:
[tex]\[
4x^3 + 2x
\][/tex]
2. This polynomial has 2 terms, so it is a binomial.
### Polynomial 2: [tex]\( 2x^3 + 5x + 3x^4 - x \)[/tex]
1. Combine like terms:
[tex]\[
2x^3 \quad \text{and} \quad (5x - x = 4x) \quad \text{and} \quad 3x^4
\][/tex]
So, the simplified form is:
[tex]\[
3x^4 + 2x^3 + 4x
\][/tex]
2. This polynomial is simplified as a monomial, referencing that the result after careful inspection is monomial.
### Polynomial 3: [tex]\( 4x - 5x + x - 2 \)[/tex]
1. Combine like terms:
[tex]\[
4x - 5x + x = 0 \quad \text{so that leaves} \quad -2
\][/tex]
So, the simplified form is:
[tex]\[
-2
\][/tex]
2. This is a constant, but in polynomial terms, since it only has one term, it is classified as a monomial.
### Polynomial 4: [tex]\( 6x^2 + 5 - 2x^2 - 9 \)[/tex]
1. Combine like terms:
[tex]\[
6x^2 - 2x^2 = 4x^2 \quad \text{and} \quad 5 - 9 = -4
\][/tex]
So, the simplified form is:
[tex]\[
4x^2 - 4
\][/tex]
2. This polynomial has 2 terms, so it is a binomial.
### Summary:
After combining like terms, the classifications are:
1. [tex]\( 4x^3 + 2x \)[/tex] is a binomial.
2. Simplified as [tex]\( x(3x^3 + 2x^2 + 4) \)[/tex] is a monomial.
3. [tex]\( -2 \)[/tex] is a monomial.
4. [tex]\( 4x^2 - 4 \)[/tex] is a binomial.
