Video: Finding the Set of Zeros of a Cubic Function

Find the set of zeros of the function 𝑓(π‘₯) = 7π‘₯(π‘₯ βˆ’ 1)(π‘₯ + 6).

02:52

Video Transcript

Find the set of zeros of the function 𝑓 of π‘₯ equals seven π‘₯ times π‘₯ minus one times π‘₯ plus six.

So first of all, what do we mean by the set of zeros of the function? Well, the zeros of the function 𝑓 of π‘₯ are the values of π‘₯ for which 𝑓 of π‘₯ is equal to zero. So to find the set of zeros of the function, we need to solve 𝑓 of π‘₯ equals zero, and then put those solutions into a set. So we write down 𝑓 of π‘₯ equals zero, the equation we want to solve. We use the definition of 𝑓 of π‘₯, so we have that seven π‘₯ times π‘₯ minus one times π‘₯ plus six is equal to zero. We’re given that the product of these three factors seven π‘₯, π‘₯ minus one, and π‘₯ plus six is equal to zero. And the only way that that can happen is if one of those factors itself is equal to zero. So either seven π‘₯ is equal to zero or π‘₯ minus one is equal to zero or π‘₯ plus six is equal to zero.

The law or property that allows us to say that one of the factors must be zero is called the zero product property. Anyway, in the place of one difficult equation, we now have three simple equations which we can solve individually. For the first equation, that seven π‘₯ equal zero, we divide by seven on both sides. And so we find that, in this case, π‘₯ is equal to zero.

Now we look at the second equation, π‘₯ minus one equals zero, and we add one to both sides to get that π‘₯ is equal to one. And we subtract six from both sides of our final equation to get that π‘₯ is equal to negative six. Looking back at the equation that we wish to solve, seven π‘₯ times π‘₯ minus one times π‘₯ plus six equals zero, we can see that when π‘₯ is zero, the first factor of the left-hand side, seven π‘₯, is equal to zero. And so the whole expression on the left-hand side is equal to zero. When π‘₯ is equal to one, the second factor of the left-hand side will be equal to zero, and so the whole expression on the left-hand side will be equal to zero. And when π‘₯ is equal to negative six, the third factor of the left-hand side, π‘₯ plus six, will be equal to zero, and so the whole expression on the left-hand side will be equal to zero. The zeros of the function are therefore zero, one, and negative six. These are the three values which make the function zero. And we want the set of zeros so we need to put these values in a set. Of course, the order that they’re put in that set doesn’t matter.

The set of zeros of our function is therefore the set containing zero, negative six, and one.

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