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

Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.

$ƒ (x) = 3 x^{4} + 2 x^{3} - 4 x^{2} + x - 1$ ; no real zero less than -2

Accepted Answer

Hey, everyone in this problem for the following polynomial function, we're asked to determine whether the real neuro satisfies the given condition or not. Now, we're given the function F of X is equal to two X to the exponent four plus three X cubed minus six X squared plus seven. And we're told that no real zero is le there is no real zero less than negative four. So we have two answer choices here. Yes, that statement is true or the, the real zero satisfies this condition or no, it doesn't satisfy this condition. Now, what we're gonna do here is make use of this really neat, the called the lower bound zero. And sometimes it's also called the bounded this theorem. So you may have seen either of these terms in your course or your textbook. And what this tells us is that if we take our function F of X and we divide it by some number, say C OK, where C is negative using synthetic division. And in the bottom row of our synthetic division table, that bottom row of values that we get if they alternate signs from positive, negative, positive, negative, positive, negative then that value of C that we divided by is a lower bound for the real zero. So this statement that there is no real zero, less than negative four means that we have a lower bound of negative four on our real zeros. So what we wanna do is take our function F divided by negative four using synthetic division and see what we get. We're gonna set up our synthetic division table on the left hand side, we have negative four. That value we're interested in for our lower bound. And then we're gonna set up the top row and our table has two rows within it and then one row underneath. So that first row in our table is gonna be the coefficients corresponding to our function F of X. We're gonna start with the highest exponent term which is X to the exponent four and it has a coefficient of two. We move to our next lowest exponent term X cubed coefficient of three. Next lowest exponent term X squared coefficient negative six. Next lowest exponent term will be an X term. Uh Now there is no X term written in our function F FX, which means that that coefficient is actually zero. OK. So just because it's not written in our function doesn't mean we can ignore it in our synthetic division table. It's really important to include the zeros if we have a coefficient of zero. OK. So we do have to include that zero for our X term. And then we move to our constant term, which is seven. OK? So that first row from left to right, 23, negative 607. Now we're gonna perform the division. We're gonna start on the left. We're gonna bring this two down and underneath our table. When we have a value under our table, we multiply it to the value on the left of our table. Two multiplied by negative four gives us negative eight. We're gonna write that in the next column over in the second row of the table. Now we have a complete column in this second column. We're gonna add within that column three plus negative eight that gives us negative five. And we write it down below a new value down below. We multiply it to the value on the left negative five multiplied by negative four gives us 20. We write it in the next column over in the second row. And then we add within the column negative six plus gives us 14. Write it down below new value down below. We multiply it to the value on the left 14 multiplied by negative four. Gives us negative 56. We write it in the column over second row and then we add within that column zero plus negative 56 gives us negative 56. We write it down below a new number down below. Now we multiply it to the number on the left negative 56 multiplied by negative four gives us 224. We write it in the next column over in the second row. And we're gonna add. Now, in this final column, we have seven plus 224 which gives us 231. Now, if we look at this last row in our table, you'll notice that the signs alternate. We have positive two negative five, positive 14, negative 56 positive 231 positive, negative, positive, negative positive. So we have the designs alternate. Oops, they alternate. OK. And what this tells us is that our lower bound theorem. OK. So using our lower bound theorem, this tells us that negative four is a lower amount on the real zeroes. If negative four is a lower bound on the real zeros, we can also say that there will be no real zero, less than negative four. And that's the wording that was used in this statement. That means that the statement we were given is true. So the real zeros will satisfy this statement. So we get answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.
$ƒ (x) = 3 x^{4} + 2 x^{3} - 4 x^{2} + x - 1$ ; no real zero less than -2

Key Concepts

Real Zeros of Polynomial Functions

Evaluating Polynomial Values to Test Inequalities

Intermediate Value Theorem

Watch next