Question Video: Finding the Set of Zeros of a Polynomial Function Mathematics

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.