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)=3x^3−8x^2+x+2; between 2 and 3

Accepted Answer

Welcome back. I'm so glad you're here. We are given the function F of X equals five X cubed minus three X squared plus two X plus one. And we're told that it has a real zero in between negative one and zero. We are to prove this. Using the intermediate value theorem. We recall from previous lessons that the intermediate value theorem tells us that if F of negative one and F of zero have opposite signs then at least one real zero will lie in between them. So what do we need to do? We need to test out F of negative one and we need to test out F of zero. See if they have opposite signs. Alright, so for F of negative one we're going to fill in a negative one everywhere that F of X had an X. So it'll be five times a negative one cubed minus three times a negative one squared plus two times a negative one plus one. Let's do the Exponents -1. Cute is still negative one minus three times negative one squared. That is going to be a positive one plus two times negative one plus one. Now let's do the multiplication. Five times negative one is negative five negative three times one is negative +32 times negative one is negative two and then plus one. So we've got negative five minus three, negative eight minus two. More. Negative 10 plus one is negative nine. So F of negative one is a negative nine. And the important part here is to note that this is a negative number. Now we are going to put a zero in for our X. And we're going to see if it changes the sign. So we've got five times zero cubed minus three times zero squared plus two times zero plus one. Okay, well, these first three those are all going to be zero when multiplying by a zero. So we're left with a positive one. F of zero equals one. And that is a positive number. And yes, we do have a sign change. Therefore, the intermediate value theorem tells us that we do have a real zero in between negative one and zero. We look at our answer choices and this matches with answer choice B. Well done. We'll catch 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)=3x^3−8x^2+x+2; between 2 and 3