Video Transcript
Find the missing constant 𝑘 if the
degree of the binomial 𝑥𝑦 squared 𝑧 raised to the power of 𝑘 plus three 𝑥
squared is seven.
In this question, we are given a
binomial in which one of the variables is raised to an unknown constant exponent of
𝑘. We need to use the fact that the
binomial has degree seven to find the value of 𝑘.
To answer this question, we can
begin by recalling that the degree of a polynomial is the maximum sum of the
exponents of all of the variables in a single term. Therefore, to find the degree of a
polynomial, we need to find the sum of the exponents of the variables in each
term.
Let’s start with the first
term. We know that 𝑥 is the same as 𝑥
raised to the first power. So the first term can be rewritten
as 𝑥 raised to the first power times 𝑦 squared times 𝑧 raised to the power of
𝑘. We then need to find the sum of the
exponents of the variables in this term. This gives us one plus two plus
𝑘. This is also the degree of this
monomial term.
In the second term of the binomial,
we only have a single variable of 𝑥. We can then recall that in cases
like this, we say that the degree of this term is just the exponent of the variable,
which is two. Since we know that the degree of
the binomial is seven, the larger of the degrees of the terms must be equal to
seven.
We know that one plus two plus 𝑘
is greater than two. So we must have one plus two plus
𝑘 equals seven. We can then solve this equation for
𝑘. We evaluate, subtract three from
both sides of the equation, to obtain that 𝑘 is equal to four.
