Show that the real zeros of each polynomial function satisfy the ...

Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^4-x^3+3x^2-8x+8; no real zero greater than 2

Accepted Answer

Hey, everyone in this problem for the following polynomial function, we're asked to determine whether the real neuros satisfy the given condition or not. We have the function F of X is equal to X to the exponent four minus three, X cubed plus six, X squared minus 12 X plus 10. And the condition we're given is that there is no real zero greater than three. For this problem, we just have two answer choices. Option A yes or option B no, now no real zero greater than three means that we have this upper bound on our real zeros. So let's recall something called the upper bound theorem. And sometimes it's also referred to as the bounded this theorem. So you may have heard either term in your course or in your textbook. And what this theorem tells us is that if we take our function F of X and we divide it by this value that we're looking at for this upper bound. And in this case, three using synthetic division and that final rowers or synthetic division table has values that are all non negative. Then this number that we divided by is going to be an upper bound on the real zeros. OK. That's exactly what we want to look for. So what we're gonna do is we're gonna set up a synthetic division table on the left hand side of our table, we have the value three that we're looking at whether or not it's an upper bound. Our table has two ropes. And in the first row, we're gonna fill it in with the coefficients of our function F of X. On the left hand side, we're gonna have the coefficient corresponding to the highest exponent term all the way down to the right hand side where we have the coefficient just of the constant term. So the highest exponent term is X to the exponent four. It has a coefficient of one. Next, we have our X Q with the coefficient of negative three, our X squared term with a coefficient of six, our X term with a coefficient of negative 12. And finally our constant term of 10. OK. So from left to right in that first row, one negative 36, negative 12 10. Now we're gonna perform synthetic division. So we're gonna start on the left hand side with this one. We're gonna bring it down underneath our table. When we have a value underneath our table, we multiply it to the value on the left of our table. One multiplied by three is three. We write it in the next column over in the second row. And now we add in that column negative three plus three gives us zero. We write it down below. We have a new number down below. We multiply it to the number on the left zero, multiplied by three gives us zero. We write it in the next column over in the second row. And then we add in that column six plus zero gives us six. We write it down below and then we multiply that number below to the number on the left hand side six multiplied by three gives us 18. We write it in the next column over in the second row. And then we add in the column negative 12 plus gives us six. We write it down below. And again, we multiply to the number on the left six multiplied by three gives us 18. We write it in the next column over in the second row and we add in that column. And this time this is our final column. We have 10 plus 18, which gives us 28. Now we wanna look at this bottom rope of our synthetic division table. We have 1066 and 28. All of these values are non negative and we have all positives and we have a zero, a zero is non negative. We just want values that aren't negative. So it's OK to have that zero, all our values are non negative. So our upper bound theorem tells us that that number we divided by which is three in this case is an upper bound or the real zeroes. All right. So if three is an upper bound for the real zeros, that means that there is no real zero, greater than three, that means the condition or the statement we were given is true. It is satisfied by this function. And so we have answer choice. A yes. Thanks everyone for watching. I hope this video helped see you in the next one.

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Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^4-x^3+3x^2-8x+8; no real zero greater than 2

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