### Video Transcript

Consider this list of
expressions. (A) Negative one minus 𝑥 plus
one. (B) 𝑥 plus 𝑦. (C) Two 𝑥𝑦 minus three 𝑥 to the
fourth power plus one. (D) 𝑥 squared plus 𝑥 to the power
of one-third plus two 𝑥. (E) Negative three 𝑥 plus two 𝑥
plus zero. Which expression, or expressions,
is a trinomial?

In this question, we are given a
list of five algebraic expressions and asked to determine which of these expressions
is a trinomial. We can do this by recalling that
for an expression to be a trinomial, it must be both a polynomial and it must have
three terms after simplification. This means that we need to check if
each of these expressions is a polynomial and if each expression has three terms
after simplification.

To do this, we can start by
recalling that a polynomial is the sum of monomials, which in turn are the product
of constants and variables where the variables must be raised to nonnegative integer
exponents. We allow for constant terms in
polynomials since raising 𝑥 to an exponent of zero is equivalent to one. We can use this definition to check
if each expression is a polynomial by checking if every term in each expression is
the product of constants and variables raised to nonnegative integer exponents.

In expression (A), we see that
there are two constant terms and only one term containing a variable. We can recall that 𝑥 is the same
as 𝑥 to the first power. So the variables in this term are
all raised to nonnegative integer exponents. Since every term in this expression
is a monomial, we have shown that expression (A) is a polynomial.

We can follow the exact same
process for expressions (B), (C), and (E). We can write a variable such as 𝑥
or 𝑦 as 𝑥 to the first power or 𝑦 to the first power and note that every term is
the product of constants and variables raised to nonnegative integer exponents. So these are all polynomials.

If we try and apply this process to
expression (D), we can note that the variable 𝑥 in the second term has an exponent
of one-third. This is not an integer, so this
expression is not a polynomial. This in turn also means that this
expression cannot be a trinomial.

Now that we have shown that the
four remaining expressions are polynomials, we need to determine which expressions
have three terms after simplification. It is important to check if we can
simplify the expressions. For instance, in expression (A), we
can note that negative one plus one is equal to zero. So expression (A) simplifies to
give negative 𝑥. Since this expression is a single
term and we have shown that it is a polynomial, we can say that this is a monomial,
not a trinomial.

In expression (B), we note that
there are only two terms. So this is a binomial and not a
trinomial. In expression (C), we cannot
simplify the expression, and we can see that there are three terms. Since we have shown that this is
also a polynomial, it must be a trinomial. Finally, in expression (E), we can
note that adding zero will not affect the polynomial. So we can remove this. And then we see that the expression
does not have three terms, so it is not a trinomial.

Therefore, of the five given
algebraic expressions, we were able to show that only expression (C) is a
trinomial.