To determine which algebraic expressions are binomials, we need to understand the definition of a binomial. A binomial is an algebraic expression that consists of exactly two terms joined by a plus (+) or minus (−) sign.
Let's examine each of the given expressions to check if they fit this definition:
1. Expression: [tex]\( x y^{\sqrt{8}} \)[/tex]
- This expression consists of a single term: [tex]\( x y^{\sqrt{8}} \)[/tex]. Therefore, it is not a binomial.
2. Expression: [tex]\( x^2 y - 3 x \)[/tex]
- This expression consists of two terms: [tex]\( x^2 y \)[/tex] and [tex]\( -3 x \)[/tex]. Therefore, it is a binomial.
3. Expression: [tex]\( 6 y^2 - y \)[/tex]
- This expression consists of two terms: [tex]\( 6 y^2 \)[/tex] and [tex]\( -y \)[/tex]. Therefore, it is a binomial.
4. Expression: [tex]\( y^2 + \sqrt{y} \)[/tex]
- This expression consists of two terms: [tex]\( y^2 \)[/tex] and [tex]\( \sqrt{y} \)[/tex]. Therefore, it is a binomial.
5. Expression: [tex]\( 4 x y - \frac{2}{5} \)[/tex]
- This expression consists of two terms: [tex]\( 4 x y \)[/tex] and [tex]\( -\frac{2}{5} \)[/tex]. Therefore, it is a binomial.
6. Expression: [tex]\( x^2 + \frac{3}{x} \)[/tex]
- This expression consists of two terms: [tex]\( x^2 \)[/tex] and [tex]\( \frac{3}{x} \)[/tex]. Therefore, it is a binomial.
Based on our examination, the following expressions are binomials:
- [tex]\( x^2 y - 3 x \)[/tex]
- [tex]\( 6 y^2 - y \)[/tex]
- [tex]\( y^2 + \sqrt{y} \)[/tex]
- [tex]\( 4 x y - \frac{2}{5} \)[/tex]
- [tex]\( x^2 + \frac{3}{x} \)[/tex]
Thus, all the given expressions except the first one, [tex]\( x y^{\sqrt{8}} \)[/tex], are binomials. The expressions that are binomials are:
[tex]\[ \boxed{2, 3, 4, 5, 6} \][/tex]
