Let's analyze and classify each polynomial by combining like terms first, and then determining if each resulting polynomial is a monomial, binomial, or trinomial.
1. Polynomial: [tex]\( x^3 + 3x^3 + 2x \)[/tex]
- Combine like terms: [tex]\( (x^3 + 3x^3) + 2x = 4x^3 + 2x \)[/tex]
- This polynomial now has two terms, [tex]\( 4x^3 \)[/tex] and [tex]\( 2x \)[/tex].
- Classification: A polynomial with two terms is a binomial.
2. Polynomial: [tex]\( 2x^3 + 5x + 3x^4 - x \)[/tex]
- Combine like terms: [tex]\( 3x^4 + 2x^3 + (5x - x) = 3x^4 + 2x^3 + 4x \)[/tex]
- This polynomial now has three terms, [tex]\( 3x^4 \)[/tex], [tex]\( 2x^3 \)[/tex], and [tex]\( 4x \)[/tex].
- Classification: A polynomial with three terms is a trinomial.
3. Polynomial: [tex]\( 4x - 5x + x - 2 \)[/tex]
- Combine like terms: [tex]\( (4x - 5x + x) - 2 = 0x - 2 = -2 \)[/tex]
- After combining like terms, this simplifies to [tex]\(-2\)[/tex], which is just one term.
- Classification: A polynomial with one term is a monomial.
4. Polynomial: [tex]\( 6x^2 + 5 - 2x^2 - 9 \)[/tex]
- Combine like terms: [tex]\( (6x^2 - 2x^2) + (5 - 9) = 4x^2 - 4 \)[/tex]
- This polynomial now has two terms, [tex]\( 4x^2 \)[/tex] and [tex]\(-4 \)[/tex].
- Classification: A polynomial with two terms is a binomial.
So, the classifications are:
- [tex]\( x^3 + 3x^3 + 2x \)[/tex] is a binomial.
- [tex]\( 2x^3 + 5x + 3x^4 - x \)[/tex] is a trinomial.
- [tex]\( 4x - 5x + x - 2 \)[/tex] is a monomial.
- [tex]\( 6x^2 + 5 - 2x^2 - 9 \)[/tex] is a binomial.
These classifications are:
[tex]$
\begin{array}{l}
\text{Binomial} \\
\text{Trinomial} \\
\text{Monomial} \\
\text{Binomial}
\end{array}
$[/tex]
