Which of the following are binomials?

A. [tex]x^2+3[/tex]
B. [tex]x^{11}[/tex]
C. [tex]x^4+x^2+1[/tex]
D. [tex]8x[/tex]
E. [tex]6x^2+\frac{1}{2} y^3[/tex]
F. [tex]\frac{5}{7} y^3+5 y^2+y[/tex]



Answer :

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.