 In this video, we provide the solution to question number 13 for practice exam number three for math 1050. We're given a polynomial f is 2x to the 6 minus 18x to the 4th plus x squared minus 9. Notice that the fifth, third, and first powers are missing from that polynomial. We're supposed to then use synthetic division where we divide f by x minus c, where in this case, c is negative 3. And then we'll analyze that. But let's first do the synthetic division. So there's a space below to do exactly that. Write down the coefficients of f in descending order. Make sure you don't skip over any places. So we have the x squared. We have 0x to the fifth. It's important to remember that. Negative 18x to the 4th, 0x cubed, 1x squared, 0x and then negative 9 like so. And then we're dividing by negative 3. I should say we're dividing by x minus negative 3, so x to plus 3. But you put the intercept here, the root, which in this case is negative 3. So let's proceed to do the calculation. Drop down the 2. So you get a 2 right there. 2 times negative 3 is negative 6 plus 0 is negative 6 times 2, excuse me, times negative 3 is going to be positive 18 minus 18 is 0. That's kind of fun. 0 times negative 3 is 0 plus 0 is 0, okay? Times negative 3 is still 0. Plus 1, we're going to get a 1 this time. 1 times negative 3 is negative 3 plus 0 is negative 3. Negative 3 times negative 3 is a positive 9. And then negative 9 plus 9 is 0. That then gives us the remainder. So the first question now that we've done this synthetic division, what's C a root? Well if the remainder is 0, you have a root. If the remainder is not 0, then we don't have a root. So in this case, since the remainder is 0, we can then say definitely that C is in fact a root. Okay, that's great. So then the next part, is C a lower bound? Is it an upper bound or is it neither? Now since C is a negative number, being an upper bound is not possible. Only positive Cs can be upper bounds and negatives could be lower bounds. Of course, the answer could be neither, whichever, but since C is negative, I know it can't be an upper bound. Let's check. Now if you're trying to be a lower bound for a negative number, you should see an alternation of sides always, plus negative, plus negative, plus negative. Now be careful on this one, zeros count as wild cards. So this is a positive, this is a negative. The next one's a zero, so yeah, you can have a positive zero. The next one is a zero again, so that can be negative, again zero is a wild card in that case. So we have positive, negative, positive, negative. The next one is then one, which is positive. The next one is then negative three, and then again zero in this case would be, it's whatever he wants, positive. So we see that alternation of sign, positive, negative, positive, negative, positive, even with the zeros, again zeros act as wilds in this case. So since we see that alternating sign each time, positive, negative, positive, negative, that does tell us that C is in fact a lower bound. So the correct answer after we've done the synthetic division is that C is a root and it's a lower bound. So any other roots of this polynomial would have to be larger than negative three.