Video Transcript
Simplify five 𝑥 minus two over
three 𝑥 squared minus seven 𝑥 minus two over nine 𝑥 to the power of four.
For this question, we have been
asked to simplify the difference of two rational expressions. Remember that a rational expression
is the quotient of two polynomials. We can think of these expressions
as algebraic fractions. And hence when trying to simplify
sums and differences of rational expressions, there are many similarities to the
methods used for numerical fractions.
For instance, the first step that
we’ll need to perform is expressing both of our terms with the same denominator. One way to do this is to multiply
the top and bottom of our first term by the denominator of our second term, which is
nine 𝑥 to the power of four. We then multiply the top and bottom
of our second term by the denominator of our first term, which is three 𝑥
squared. Doing so would give us a common
denominator of 27𝑥 to the power of six for both of our terms. Although this would be a perfectly
reasonable method, we can actually cut down on our calculations by first considering
whether our denominators have a common factor. Let’s clear some room and take a
step back.
Looking at our two denominators, we
can see that nine 𝑥 to the power of four is equal to three 𝑥 squared times three
𝑥 squared. This means that yes, our two
denominators do share a common factor of three 𝑥 squared. If we multiply the top and bottom
of our first term by three 𝑥 squared, it will match the denominator of our second
term. After doing so, both terms will
share the same denominator of nine 𝑥 to the power of four. And this is exactly the condition
that we need to continue with our simplification. Five 𝑥 minus two multiplied by
three 𝑥 squared is 15𝑥 to the power of three minus six 𝑥 squared. This is the numerator of our first
term. The denominator is, of course, nine
𝑥 to the power of four.
Now that both of our terms have the
same denominator, we can express the given difference as a single algebraic
fraction. Note that the final term of our
numerator has a positive sign. Since we are subtracting a negative
two, we are left with a positive two. In the resulting algebraic
fraction, we cannot combine any of the terms in our numerator, nor can we cancel any
common factors. Since there are no more useful
simplifications, this is our answer. The simplified version of the given
expression is 15𝑥 cubed minus six 𝑥 squared minus seven 𝑥 add two over nine 𝑥 to
the power of four.
As a final note, our question did
not ask us to consider this rational expression as a function. If it had, we might need to include
extra steps early on in our method to consider the domain of the function. Considering the domain is not
relevant to this question, but it is definitely worth looking out for in the
future.
