>> The next technique we'll talk about is called the rational zero test, and it works like this. Suppose we have a function like this. It turns out that just by observing and making a little calculation on the leading coefficient and the constant, we can determine all of the possible rational zeros. Now here's how the calculation is made, and I'll show you how it's used in just a minute. But the all of the possible rational zeros can be calculated by writing all of the fractions possible using the factors of the constant in the numerator of this fraction, and the factors of the leading coefficient as denominators of the fraction. So we have, let's see, factors of the, of the constant, the constant is 10. So all the factors of the constant would be plus or minus 1, plus or minus 2, plus or minus 5, plus or minus 10. And now in the denominator we have factors of the leading coefficient. The leading coefficient was 5, so all of the possible factors, plus or minus 1, plus or minus 5. Now we have a little list to go through here. We have all these guys over 1, and then all of these guys over 5. That's the complete list. Well we put all of these over 1, and we just have all of these items. So we have plus or minus 1, plus or minus 2, plus or minus 5, plus or minus 10. Then we go through all of these over 5. Well let's see, plus or minus 1 over 5, plus or minus 2 over 5, and then plus or minus 5 over 5 is just 1, we already have it in there, plus or minus 10 over 5 is plus or minus 2, and we already have it in there. Now we have this list of all of the possible rational zeros. And you might be thinking, well why do we want to go through all of this trouble to identify possible rational zeros? Well this is a, a test. This is a, a technique that you would use when your sort of bump into a need for it. And here's how you might bump into that need. Let's say that we begin with this problem to build a table of values. You know, here's our function, and let's, let's build a little simple table of values just right up here. We're, we're gonna just start with an X and Y, XY pairs, and let's just say X is zero. Now if X is zero, all of these terms go out and we have 10. So when X is zero, Y is 10. Now what about when X is 1? If X is 1 then we have 5 minus 2 minus 25 plus 10. Let's see, 5 minus 2 is 3, 3 minus 25 is negative 22, plus 10 is negative 12. So when X is 1, Y is negative 12. Let's put these points onto a coordinate plane. Now just relatively speaking, zero goes to 10 is just up here somewhere. And 1 negative 12 is down here somewhere. Now with the intermediate value theorem we know that there is a crossing in the interval from zero to 1. So we have a zero in here. And, you know, zeros, if we can identify a zero, it can lead us to a factor, and that will kind of make the, the function a little more, a little easier to, to deal with. All right, so we're, we are hoping that that crossing is rational. Now to test whether or not we have a rational value for the, the zero of the function, we can use the rational zero test to determine which items we test for. You see? There are an infinite number of rational values between zero and 1, which ones do we test for? Well we, we use our rational zero test and we realize that between zero and 1 the values in that list that are possible would be simply 1/5 and 2/5's. So that's all we need to try. All right, so let's go through and try those. We would have, synthetically now, we would divide by 1/5. And so bring down the 5. 1/5 times 5 is 1, negative 2 with 1 is negative 1, 1/5 times negative 1 is negative 1/5. Uh oh, together we get negative 25 and 1/5. Uh oh. 1/5 times negative 25 and 1/5 is a mess, and when you add 10 to this business you certainly don't get negative 10 here, you see, to collect to get zero. So this value is not zero, that's the point here. All right, so 1/5 is not a zero of the function. Now let's try 2/5's. Bring down the 5. 2/5's times 5 is 2, 2 with negative 2 is zero, that's nice, 2/5's times zero is zero, together negative 25. 2/5's times negative 25, the 5 and the negative 25 cancel leaving negative 5. 2 times negative 5, negative 10, 10 with negative 10, zero. So we do indeed have a zero at 2/5's.