To determine which algebraic expression is a trinomial, we need to understand what a trinomial is. A trinomial is a polynomial with exactly three terms. Each term must be separated by a plus (+) or minus (-) sign.
Let's analyze each given expression to see if it is a trinomial:
1. Expression 1: [tex]\( x^3 + x^2 - \sqrt{x} \)[/tex]
- [tex]\( x^3 \)[/tex] is one term.
- [tex]\( x^2 \)[/tex] is another term.
- [tex]\( -\sqrt{x} \)[/tex] is the third term.
This expression has three terms, so it is a trinomial.
2. Expression 2: [tex]\( 2x^3 - x^2 \)[/tex]
- [tex]\( 2x^3 \)[/tex] is one term.
- [tex]\( -x^2 \)[/tex] is another term.
This expression has only two terms, so it is not a trinomial.
3. Expression 3: [tex]\( 4x^3 + x^2 - \frac{1}{x} \)[/tex]
- [tex]\( 4x^3 \)[/tex] is one term.
- [tex]\( x^2 \)[/tex] is another term.
- [tex]\( -\frac{1}{x} \)[/tex] is the third term.
This expression has three terms, so it is a trinomial.
4. Expression 4: [tex]\( x^6 - x + \sqrt{6} \)[/tex]
- [tex]\( x^6 \)[/tex] is one term.
- [tex]\( -x \)[/tex] is another term.
- [tex]\( \sqrt{6} \)[/tex] is the third term.
This expression has three terms, so it is a trinomial.
After reviewing all the expressions, we find that expressions 1, 3, and 4 are trinomials. However, based on the provided analysis, the first expression [tex]\( x^3 + x^2 - \sqrt{x} \)[/tex] matches the criteria for a trinomial.
Thus, the expression that is a trinomial is:
[tex]\[ x^3 + x^2 - \sqrt{x} \][/tex]
