Find the set of zeros of the function 𝑓 of 𝑥 equals 𝑥 cubed plus five 𝑥 squared minus nine 𝑥 minus 45.
This is a polynomial function. In particular, it’s a cubic function. And we’re going to find the zeros of this function by attempting to factorise the cubic expression that we have. We can separate the terms of this cubic expression into two groups. One of which is 𝑥 cubed plus five 𝑥 squared, and the other is minus nine 𝑥 minus 45. We can notice that both groups have a factor of 𝑥 plus five. 𝑥 cubed plus five 𝑥 squared is 𝑥 squared times 𝑥 plus five, and negative nine 𝑥 minus 45 is equal to negative nine times 𝑥 plus five.
Now that we have two things with a common factor, we can combine them to get 𝑥 squared minus nine times 𝑥 plus five. So we have factored 𝑓 of 𝑥 somewhat; we’ve written it as a product of two factors. But we notice the factor of 𝑥 squared minus nine is a difference of two squares and so itself can be factored. 𝑥 squared minus nine is equal to 𝑥 plus three times 𝑥 minus three. And so 𝑓 of 𝑥 is equal to 𝑥 plus three times 𝑥 minus three times 𝑥 plus five.
Now that we have completely factored 𝑓 of 𝑥, we can find its set of zeros. We wanted to find the set of values of 𝑥 for which 𝑓 of 𝑥 is equal to zero. Using the factored form of 𝑓 of 𝑥, we get that 𝑥 plus three times 𝑥 minus three times 𝑥 plus five is equal to zero. We have that the product of three numbers is zero, and the only way that that can happen is if one of those numbers is zero. So either 𝑥 plus three is equal to zero, or 𝑥 minus three is equal to zero, or 𝑥 plus five is equal to zero. So 𝑥 is equal to negative three, or 𝑥 is equal to three, or 𝑥 is equal to negative five.
In the question, we’re asked for the set of zeros. So we need to put these three values into a set. 𝑥 is in the set that contains negative five, three, and negative three. The order that we write the elements of the set doesn’t matter. If 𝑥 is a zero of the function 𝑓 of 𝑥, then it’s either negative five or three or negative three.
And so the set of zeros of 𝑓 of 𝑥, which is after all what we’re looking for, is the set of negative five, three, and negative three.