To determine which of the given expressions are binomials, we need to understand what a binomial is. A binomial is a polynomial with exactly two terms.
Let's examine each option:
A. [tex]\( x^2 + 3 \)[/tex]:
This has two terms ([tex]\( x^2 \)[/tex] and 3). Therefore, it is a binomial.
B. [tex]\( 6x^2 + \frac{1}{2}y^3 \)[/tex]:
This has two terms ([tex]\( 6x^2 \)[/tex] and [tex]\( \frac{1}{2}y^3 \)[/tex]). Therefore, it is a binomial.
C. [tex]\( 8x \)[/tex]:
This has only one term (8x). Therefore, it is not a binomial.
D. [tex]\( x^4 + x^2 + 1 \)[/tex]:
This has three terms ([tex]\( x^4 \)[/tex], [tex]\( x^2 \)[/tex], and 1). Therefore, it is not a binomial.
E. [tex]\( \frac{5}{7}y^3 + 5y^2 + y \)[/tex]:
This has three terms ([tex]\( \frac{5}{7}y^3 \)[/tex], [tex]\( 5y^2 \)[/tex], and y). Therefore, it is not a binomial.
F. [tex]\( x^{11} \)[/tex]:
This has only one term ([tex]\( x^{11} \)[/tex]). Therefore, it is not a binomial.
Conclusion:
The expressions that are binomials are:
- A. [tex]\( x^2 + 3 \)[/tex]
- B. [tex]\( 6x^2 + \frac{1}{2}y^3 \)[/tex]
Hence, the indices of the binomials are 0 and 2, which correspond to:
- A. [tex]\( x^2 + 3 \)[/tex]
- C. [tex]\( 8x \)[/tex]
