To determine which of the provided options is a trinomial that includes a constant term, we need to analyze each expression for both its structure (tri means three, so it must have three terms) and the presence of a constant term (a term without any variables). Let's break down each option:
A. [tex]\( x \)[/tex]
- This expression has only one term, which is [tex]\( x \)[/tex].
- [tex]\( x \)[/tex] is not a trinomial, as it does not have three terms.
- There is no constant term in this expression.
B. [tex]\( x + 2y + 10 \)[/tex]
- This expression has three terms: [tex]\( x \)[/tex], [tex]\( 2y \)[/tex], and [tex]\( 10 \)[/tex].
- Since there are three terms, it is indeed a trinomial.
- The term [tex]\( 10 \)[/tex] is a constant term because it does not contain any variables.
C. [tex]\( y^6 + 8y^3 + 64y \)[/tex]
- This expression has three terms: [tex]\( y^6 \)[/tex], [tex]\( 8y^3 \)[/tex], and [tex]\( 64y \)[/tex].
- While it is a trinomial, none of the terms are constant. Each term includes the variable [tex]\( y \)[/tex].
D. [tex]\( x^3 + y \)[/tex]
- This expression consists of two terms: [tex]\( x^3 \)[/tex] and [tex]\( y \)[/tex].
- This is not a trinomial, as it does not have three terms.
- There is no constant term in this expression.
Based on this analysis, the only option that fits the criteria of being a trinomial with a constant term is:
B. [tex]\( x + 2y + 10 \)[/tex]
Therefore, the correct answer is B.
