In Exercises 33–40, use the Intermediate Value Theorem to show th...

Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^4+6x^3−18x^2; between 2 and 3

Accepted Answer

Welcome back. I'm so glad you're here. We are given the polynomial function F of X equals to X to the fourth plus five X to the third minus nine X squared. And we're told that it has a real zero in between one and two. We are to prove this using the intermediate value theorem. Well, we recall from previous lessons that the intermediate value theorem tells us that if F. Of one and F of two have opposite signs, then at least one real zero lies in between one and two. So let's test that by finding out what half of one is. Then we'll compare its sign to F of two. So for F of one we're going to plug a one in anywhere. There's an X in our original F of X function. So we'll have two times one to the fourth plus five times +12. The third minus nine times one squared. Well, we'll start off with the exponents but that's not going to be too difficult because well, one to the fourth is just one. So two times one plus five times one of the third is just one. So five times one minus nine times one square. That's just one. Now we'll do the multiplication, two times one is two plus five times one is five minus nine times one is minus nine. Well two plus five is seven minus nine gives us a negative two. So F of one equals negative two and note that that is a negative answer. So now we are going to compare that with F of two. We'll see if we get a positive answer for F. Of two. We're going to plug two in for every X. In our original affects equation. So we've got two times two to the fourth plus five times two to the third minus nine times two squared. Let's do the exponents first, so two times two to the fourth to the fourth is plus five times two to the third, Two to the third is eight minus nine times two squared two squared is four, two times 16 is 32 plus five times eight is 40 minus nine times four is minus 36. We add and subtract all those numbers and we'll get an answer of 36. So F of two is 36. And note that yes, this is a positive answer. So yes. The intermediate value theorem tells us that there is a real zero in between one and two because F of one and F of two gave us a sign change. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one

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In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^4+6x^3−18x^2; between 2 and 3