Video Transcript
Find the set of zeros of the
function 𝑓 of 𝑥 equals 𝑥 to the fourth power minus 17𝑥 squared plus 16.
Remember, the zeros of a polynomial
function are the values of 𝑥 that make that function equal to zero. And so to find the zeros of our
function, we need to solve the equation 𝑥 to the fourth power minus 17𝑥 squared
plus 16 equals zero. And one technique we have to find
the set of zeros is to factor the expression.
So how do we factor 𝑥 to the
fourth power minus 17𝑥 squared plus 16? Well, it’s a little bit tricky to
spot, but this does look a little bit like a quadratic function. We’re going to perform a
substitution. And we’re going to let 𝑦 be equal
to 𝑥 squared. And then if we consider 𝑥 to the
fourth power as being equal to 𝑥 squared squared, 𝑥 to the fourth power can be
written then as 𝑦 squared. Similarly, negative 17𝑥 squared
can be written as negative 17𝑦. So our equation becomes 𝑦 squared
minus 17𝑦 plus 16 equals zero.
We now have a rather nice-looking
quadratic that we can solve by factoring. We’re going to have two binomials
at the front of which we must have a 𝑦. We then need to find two numbers
that have a product, they multiply to make 16, and a sum of or they add to make
negative 17. Those numbers are negative one and
negative 16. Remember, a negative multiplied by
a negative is a positive. So our equation is 𝑦 minus one
times 𝑦 minus 16 equals zero. And for the product of these
binomials to be equal to zero, we can say that either 𝑦 minus one must be equal to
zero or 𝑦 minus 16 must be equal to zero. And if we solve as normal, we see
that 𝑦 must be equal to one or 16.
But of course, we were trying to
find the zeros of our function. And we said those are the values of
𝑥 that make 𝑓 of 𝑥 equal to zero. So we’re going to now replace 𝑦
with our original substitution, 𝑥 squared. And so 𝑥 squared is equal to one
or 𝑥 squared is equal to 16. We’ll solve by taking the square
root of both sides of each equation remembering, of course, to take both the
positive and negative square root of one and 16. And so we get 𝑥 equals positive or
negative one and 𝑥 equals positive or negative four. Now, we want to write this using
set notation. And so we use these squiggly
brackets. The set of zeros of the function 𝑓
of 𝑥 is the set containing the elements negative four, negative one, one, and
four.
