### Video Transcript

Find the set of zeroes of the
function π of π₯ equals π₯ times π₯ squared minus 81 minus two times π₯ squared
minus 81.

We might begin by recalling that
the zeros of a polynomial function are the values of π₯ that make it equal to
zero. And so we need to set the
expression π₯ times π₯ squared minus 81 minus two times π₯ squared minus 81 equal to
zero and solve for π₯.

Now, if we were to distribute our
parentheses, weβd probably spot that we have a cubic. Thatβs a polynomial whose order is
three. And so we might be thinking that we
need to distribute the parentheses and go from there. However, if we look really
carefully, we see that the two terms share a common factor. They share a factor of π₯ squared
minus 81. And so to solve this equation,
weβre going to factor by removing that common factor of π₯ squared minus 81.

If we divide the first term by π₯
squared minus 81, that leaves us simply with π₯. And then if we divide the second
term by π₯ squared minus 81, we get negative two. And so we can see that the
expression π₯ times π₯ squared minus 81 minus two times π₯ squared minus 81 is equal
to π₯ squared minus 81 times π₯ minus two. And so if we can fully factor an
expression when finding the zeros, itβs really helpful because we now need to ask
ourselves, which values of π₯ make this expression equal to zero?

And of course, since weβre
multiplying π₯ squared minus 81 by π₯ minus two, and that gives us zero, we can say
that either π₯ squared minus 81 must be equal to zero or π₯ minus two must be equal
to zero. Weβll now solve each of these
equations in turn. Weβll solve our first equation by
adding 81 to both sides. And that tells us that π₯ squared
is equal to 81. Weβre now going to take the square
root of both sides. But we do need to be a little bit
careful. Weβll need to take both the
positive and negative square root of 81.

And so the solutions to the
equation π₯ squared minus 81 equals zero are π₯ equals nine and π₯ equals negative
nine. The other equation is a little bit
more straightforward. Weβre just going to add two to both
sides. And so we find another solution to
the equation π of π₯ equals zero and, therefore, a zero of our function to be
two. And we found the solutions to the
equation π of π₯ equals zero, but the question asks us to find the set of zeros of
the function. And so we use these squiggly
brackets to represent the set containing the elements negative nine, two, and
nine.

And thatβs the answer to our
question. The set of zeros of the function π
of π₯ equals π₯ times π₯ squared minus 81 minus two times π₯ squared minus 81
contains the elements negative nine, two, and nine. Note, of course, that we could go
back to our original function to check whether these answers are correct. We would need to substitute each
one in turn and double-check that we do indeed get an answer of zero.