To determine which algebraic expressions are binomials, we first need to understand what a binomial is. A binomial is an algebraic expression that contains exactly two unlike terms. Let's carefully analyze each given expression to see if it meets the criterion of being a binomial.
1. [tex]\( x y_{\sqrt{8}} \)[/tex]
- This expression contains only one term because there is no addition or subtraction separating any part of it.
- Thus, this is not a binomial.
2. [tex]\( x^2 y - 3 x \)[/tex]
- This expression has two unlike terms: [tex]\( x^2 y \)[/tex] and [tex]\( -3 x \)[/tex].
- Therefore, this is a binomial.
3. [tex]\( 6 y^2 - y \)[/tex]
- This expression has two unlike terms: [tex]\( 6 y^2 \)[/tex] and [tex]\( - y \)[/tex].
- Therefore, this is a binomial.
4. [tex]\( y^2 + \sqrt{y} \)[/tex]
- This expression consists of two unlike terms: [tex]\( y^2 \)[/tex] and [tex]\( \sqrt{y}\)[/tex].
- Therefore, this is a binomial.
5. [tex]\( 4 x y - \frac{2}{5} \)[/tex]
- This expression has two unlike terms: [tex]\( 4 x y \)[/tex] and [tex]\( -\frac{2}{5} \)[/tex].
- Therefore, this is a binomial.
6. [tex]\( x^2 + \frac{3}{x} \)[/tex]
- This expression has two unlike terms: [tex]\( x^2 \)[/tex] and [tex]\( \frac{3}{x} \)[/tex].
- Therefore, this is a binomial.
Thus, the binomials from the given list are:
- [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]
So, the binomials are:
[tex]\[ [2, 3, 5] \][/tex]
