### Video Transcript

Find the value of the constant 𝑘
given that six 𝑥 raised to the fifth power plus 𝑘𝑥 raised to the seventh power
all over two 𝑥 squared is equal to three 𝑥 cubed plus four 𝑥 raised to the fifth
power.

In this question, we want to find
the value of a constant 𝑘 by using a given equation involving operations on
polynomials. There are many ways of answering
this question, and we will go through two of these. The first way that we can find the
value of 𝑘 is to note that the left-hand side of the equation is the quotient of a
polynomial and a monomial. We can divide a polynomial by a
monomial by dividing each of the terms by the monomial separately and applying the
quotient rule for exponents. First, we split the division over
each term to obtain the following equation. Next, we recall that the quotient
rule for exponents tells us that if 𝑥 is nonzero, then 𝑥 raised to the power of 𝑚
over 𝑥 raised to the power of 𝑛 is equal to 𝑥 raised to the power of 𝑚 minus
𝑛.

It is worth noting that we know
that 𝑥 is nonzero since we cannot divide by zero. We can apply this to each term on
the left-hand side of the equation. First, we have six over two is
three. And we need to multiply this by 𝑥
raised to the power of the difference in the exponents, that is, five minus two. We can follow the same process for
the second term to get 𝑘 over two multiplied by 𝑥 raised to the power of seven
minus two. This expression must be equal to
three 𝑥 cubed plus four 𝑥 raised to the fifth power. We can then evaluate the exponents
to obtain the following equation.

It is worth reiterating at this
point that we are not working with an equation in the traditional sense. This is an equivalence. We want both sides of the equation
to be equal for all valid values of 𝑥, whereas in an equation, we are looking for
the values of a variable that satisfy the equation. We can now note that the first
terms of both expressions are identical. Similarly, the second term in both
expressions has the same variable factor of 𝑥 raised to the fifth power. Therefore, for the expressions to
be equivalent, the coefficients of the terms must be equal. So, 𝑘 over two must be equal to
four, which we can calculate is true only when 𝑘 equals eight.

It is worth noting that we can
answer this question without using division by instead multiplying the equation
through by two 𝑥 squared. Doing this gives us the following
equivalence, which we need to solve for 𝑘. We can simplify the expression on
the right-hand side by distributing the monomial factor and multiplying the powers
of 𝑥 by adding their exponents. We can then evaluate the
expressions in the exponents and once again note that the expressions are identical,
so we must have that 𝑘 equals eight.