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^3−x−1; between 1 and 2

Accepted Answer

Welcome back. I'm so glad you're here. We are given the polynomial function F of X equals X cubed plus four X minus three. And we're told that it has a real zero in between zero and one. We are to prove this. Using the intermediate value theorem. Will we recall from previous lessons that the intermediate value theorem tells us that if F of zero and F of one have opposite signs, then at least one real zero lies in between zero and one. Now let's test this. Let's find out what the sign of F of zero is. And then we'll check and see what the sign of F of one is. F of zero. We're going to plug zero everywhere. There was an X in our original F of X function. So this is going to be zero cubed plus four times zero minus three. Well zero cubed +20 plus four times zero that zero. So this is just negative three. F of zero is -3. And note that this is a negative value. Now let's test out half of one and see if we get a positive value. We're going to plug one in everywhere. We have an X in our original F of X function. So this is one cubed plus four times one minus +31 cubed is one plus four times one is plus four minus three, one plus four is five minus three is two. So F of one equals two and yes, this is a positive solution. So the intermediate value theorem tells us that because F of zero was negative and F of one is positive, there is a real zero in between zero and one. We look at our answer choices, and this matches with answer choice C. Well done, we'll catch on the next one.

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^3−x−1; between 1 and 2

Key Concepts

Intermediate Value Theorem

Continuous Functions

Polynomials and Their Properties

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