Video Transcript
Find the denominator of the
fraction in this equation: Three 𝑥 squared plus 11𝑥 plus eight over what is equal
to 𝑥 plus one.
Let’s call the denominator of our
fraction 𝑦 for now, where 𝑦 is going to be some function of 𝑥. Then we’re going to recall the
relationship between fractions and division. A fraction is just another way of
writing a division. So what this question is also
asking us is, what is the value of 𝑦 such that three 𝑥 squared plus 11𝑥 plus
eight divided by 𝑦 equals 𝑥 plus one? Now, we could rearrange this to
make 𝑦 the subject. Or we can quote that if 𝑎 divided
by 𝑏 is equal to 𝑐, then 𝑎 divided by 𝑐 must be equal to 𝑏. This makes a lot of sense because
if we rearrange each equation, we find that 𝑎 is equal to 𝑏 times 𝑐. We can think of 𝑏 and 𝑐 as a
factor pair of 𝑎. We can therefore say that three 𝑥
squared plus 11𝑥 plus eight divided by 𝑥 plus one must be equal to 𝑦.
But how do we work out this
division on the lef-hand side? Well, we have a number of methods,
but factorization is generally the most straightforward. We’re going to look to factor the
expression three 𝑥 squared plus 11𝑥 plus eight. There are a number of ways to do
this. One method is kind of
observation. It’s a quadratic equation, and
there are no common factors apart from one in each of our terms. And so we know we can write it as
the product of two binomials. The first term in each binomial
must be three 𝑥 and 𝑥 because three 𝑥 times 𝑥 gives us the three 𝑥 squared we
need.
And then we need to look for factor
pairs of eight bearing in mind that one of these is going to be multiplied by
three. And then once that happens, when we
add our numbers together, we’re going to get 11. Well, a factor pair we could use is
eight and one. And if we multiply three by one, we
get three. Then three plus eight is 11. For this to work, both eight and
one need to be positive. And so we factored our
expression. It’s three 𝑥 plus eight times 𝑥
plus one. And so we can now rewrite our
equation as three 𝑥 plus eight times 𝑥 plus one all over 𝑥 plus one equals
𝑦.
Now, our next step since it’s
written as a fraction is to simplify like we would any other fraction by dividing
through by a common factor. Here, we see we have a common
factor of 𝑥 plus one. 𝑥 plus one divided by 𝑥 plus one
is simply one. And so we see that 𝑦 is equal to
three 𝑥 plus eight? And since we said 𝑦 was the
denominator of our fraction, then the denominator is three 𝑥 plus eight. Now, a really quick way to check
this answer is to check that the product of 𝑥 plus one and our denominator is
indeed equal to three 𝑥 squared plus 11𝑥 plus eight. And in fact, if we multiply these
two binomials, we do indeed get three 𝑥 squared plus 11𝑥 plus eight.
