To determine which of the given expressions are binomials, we need to recall the definition of a binomial. A binomial is a polynomial that consists of exactly two terms. Each term can be a monomial (a single term like [tex]\( ax^n \)[/tex]).
Let's examine each expression one by one:
A. [tex]\( x^2 + 3 \)[/tex]
This expression has two terms: [tex]\( x^2 \)[/tex] and [tex]\( 3 \)[/tex]. Since there are exactly two terms, this is a binomial.
B. [tex]\( x^{11} \)[/tex]
This expression consists of just one term: [tex]\( x^{11} \)[/tex]. Since a binomial must have exactly two terms, this is not a binomial.
C. [tex]\( x^4 + x^2 + 1 \)[/tex]
This expression has three terms: [tex]\( x^4 \)[/tex], [tex]\( x^2 \)[/tex], and [tex]\( 1 \)[/tex]. Since it does not have exactly two terms, this is not a binomial.
D. [tex]\( 8x \)[/tex]
This expression has only one term: [tex]\( 8x \)[/tex]. Since a binomial must have exactly two terms, this is not a binomial.
E. [tex]\( 6x^2 + \frac{1}{2} y^3 \)[/tex]
This expression has two terms: [tex]\( 6x^2 \)[/tex] and [tex]\( \frac{1}{2} y^3 \)[/tex]. Since there are exactly two terms, this is a binomial.
F. [tex]\( \frac{5}{7} y^3 + 5 y^2 + y \)[/tex]
This expression has three terms: [tex]\( \frac{5}{7} y^3 \)[/tex], [tex]\( 5 y^2 \)[/tex], and [tex]\( y \)[/tex]. Since it does not have exactly two terms, this is not a binomial.
To summarize, the binomials among the given expressions are:
- [tex]\( x^2 + 3 \)[/tex] (Expression A)
- [tex]\( 6x^2 + \frac{1}{2} y^3 \)[/tex] (Expression E)
So, the expressions that are binomials are A and E.
