Video Transcript
Consider this list of expressions:
(a) 𝑥𝑦𝑧 plus 𝑥 squared. (b) 𝑥 cubed plus three 𝑥 squared
minus two 𝑥, (c) 𝑥𝑦 plus three, (d) 𝑦 cubed, and (e) 𝑎 squared 𝑏 plus 𝑏
cubed. Which expression, or expressions,
is a binomial of degree three?
In this question, we are given a
list of five different algebraic expressions. And we need to determine which of
these expressions is a binomial of degree three. Since we need to check if each
expression is a binomial and we need to find the degree of each term, let’s start by
checking which of the five expressions are binomials.
To do this, we first recall that a
binomial is a polynomial that has two terms. So, we need to check which
expressions are polynomials and which expressions have two terms. To help us keep track of all of
this information for each expression, let’s construct a table where we can list if
each expression is a polynomial, has two terms, and the degree of each
expression. We can start by checking if each
expression is a polynomial. We can recall that a polynomial is
the sum of monomial terms, which in turn are the products of constants and variables
raised to nonnegative integer exponents.
The main things to note about this
definition is that each term can have a constant factor and all of the variables
must be raised to nonnegative integer exponents. We also allow for a term to be a
constant nonzero value that does not have any variable as a factor. Therefore, we can check if each
expression is a polynomial by checking every term in the expression is in this
form.
In expression (a), we note that we
can rewrite the expression as 𝑥 to the first power, 𝑦 to the first power, 𝑧 to
the first power plus 𝑥 squared. The exponents of the variables are
all nonnegative integers. So, this is a polynomial.
We can follow a similar process for
the other four expressions, where we note that constant terms on their own are
allowed either by adding this into the definition or by using the fact that 𝑥 to
the zeroth power is one. This means that all five
expressions are polynomials. Let’s now check the number of terms
in each expression to see which of these polynomials are binomials.
We recall that a term in this case
refers to the number of monomials in the simplified expression. In the first expression, we see
that there are two distinct monomial terms separated by the addition. This means that expression (a) is a
polynomial with two terms. So, it is a binomial. Expression (b) is a polynomial with
three terms. So, it is a trinomial, not a
binomial. Both expressions (c) and (e) are
polynomials with two terms. So, they are also binomials. Expression (d) only has a single
term. It is worth noting that we could
write this as 𝑦 cubed plus zero. However, we do not count zero to be
a term since it does not affect the expression. Thus, expression (d) is a
monomial.
We now need to check the degree of
the binomial expressions. For due diligence, we will find the
degree of all five expressions, though it is not necessary to do this to answer the
question. To find these degrees, we can start
by recalling that the degree of a monomial is the sum of the exponents of its
variables. We can then recall that the degree
of a polynomial is the greatest degree of any of its monomial terms. This means that we can calculate
the degree of a polynomial by finding the degree of every term and then choosing the
largest value.
Let’s calculate the degree of all
five polynomials. In expression (a), we can add the
exponents of the variables in the first term to get three. And we note that there is only a
single exponent of two in the second term. So, the first term has degree
three, and the second term has degree two. The greatest of these values is
three, so its degree is three. This means that expression (a) is a
degree three binomial.
In expression (b), we see that all
of the terms have a single variable. And the largest exponent of these
variables is three. So, this is a degree three
trinomial. However, this expression has three
terms. So, it is a trinomial, and it is
not a binomial.
In expression (c), we see that the
first term has two variables, each with exponent one. So, the degree of this term is one
plus one, which equals two. The second term is a nonzero
constant, and so its degree is zero. The greatest of these degrees is
two. So, expression (c) is a binomial of
degree two, not three.
Expression (d) only contains a
single term, whose only variable is raised to an exponent of three. So, the degree of this monomial is
three, but it is not a binomial. Finally, for expression (e), we can
see that both terms are of degree three. So, the binomial itself also has
degree three.
If we then check our table, we see
that only expressions (a) and (e) are polynomials with two terms that are of degree
three.
