To determine which algebraic expression is a trinomial, we need to count the number of terms in each expression. A trinomial is defined as a polynomial with exactly three terms.
Let's analyze each given expression step-by-step.
1. [tex]\( x^3 + x^2 - \sqrt{x} \)[/tex]
This expression consists of three terms:
1) [tex]\( x^3 \)[/tex]
2) [tex]\( x^2 \)[/tex]
3) [tex]\( -\sqrt{x} \)[/tex]
Thus, the expression [tex]\( x^3 + x^2 - \sqrt{x} \)[/tex] has 3 terms.
2. [tex]\( 2x^3 - x^2 \)[/tex]
This expression consists of two terms:
1) [tex]\( 2x^3 \)[/tex]
2) [tex]\( -x^2 \)[/tex]
Thus, the expression [tex]\( 2x^3 - x^2 \)[/tex] has 2 terms.
3. [tex]\( 4x^3 + x^2 - \frac{1}{x} \)[/tex]
This expression consists of three terms:
1) [tex]\( 4x^3 \)[/tex]
2) [tex]\( x^2 \)[/tex]
3) [tex]\( -\frac{1}{x} \)[/tex]
Thus, the expression [tex]\( 4x^3 + x^2 - \frac{1}{x} \)[/tex] has 3 terms.
4. [tex]\( x^6 - x + \sqrt{6} \)[/tex]
This expression consists of three terms:
1) [tex]\( x^6 \)[/tex]
2) [tex]\( -x \)[/tex]
3) [tex]\( \sqrt{6} \)[/tex]
Thus, the expression [tex]\( x^6 - x + \sqrt{6} \)[/tex] has 3 terms.
From the analysis, the expressions with exactly three terms (trinomials) are:
- [tex]\( x^3 + x^2 - \sqrt{x} \)[/tex]
- [tex]\( 4x^3 + x^2 - \frac{1}{x} \)[/tex]
- [tex]\( x^6 - x + \sqrt{6} \)[/tex]
The expression that is the first trigonomial in the given list is:
[tex]\[ x^3 + x^2 - \sqrt{x} \][/tex]
Therefore, the algebraic expression [tex]\( x^3 + x^2 - \sqrt{x} \)[/tex] is the first trinomial in the given list.
