To determine which algebraic expressions are binomials, we need to check if each expression consists of exactly two terms. Let's analyze each expression one by one:
1. \( x y_{\sqrt{8}} \)
- This expression consists of a single term: \( x y^{\sqrt{8}} \).
- Since it is not composed of two distinct terms, it is not a binomial.
2. \( x^2 y - 3 x \)
- This expression has two distinct terms: \( x^2 y \) and \( -3 x \).
- So, it is a binomial.
3. \( 6 y^2 - y \)
- This expression has two distinct terms: \( 6 y^2 \) and \( -y \).
- Therefore, it is a binomial.
4. \( y^2 + \sqrt{v} \)
- This expression has two distinct terms: \( y^2 \) and \( \sqrt{v} \).
- Hence, it is a binomial.
5. \( 4 x y - \frac{2}{5} \)
- This expression has two distinct terms: \( 4 x y \) and \( -\frac{2}{5} \).
- Thus, it is a binomial.
6. \( x^2 + \frac{3}{x} \)
- This expression has two distinct terms: \( x^2 \) and \( \frac{3}{x} \).
- So, it is a binomial.
In summary, the expressions that are binomials are:
- \( x^2 y - 3 x \)
- \( 6 y^2 - y \)
- \( y^2 + \sqrt{v} \)
- \( 4 x y - \frac{2}{5} \)
- \( x^2 + \frac{3}{x} \)
So, the algebraic expressions that are binomials are:
[tex]\[ x^2 y - 3 x \][/tex]
[tex]\[ 6 y^2 - y \][/tex]
[tex]\[ y^2 + \sqrt{v} \][/tex]
[tex]\[ 4 x y - \frac{2}{5} \][/tex]
[tex]\[ x^2 + \frac{3}{x} \][/tex]
