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

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^3−10x+9; between -3 and -2

Accepted Answer

Hello. Today we are going to be using the intermediate value theorem to determine whether the given polynomial has a real root between negative five and zero. Before we jump into this problem, let's just quickly discuss what the intermediate value theorem is telling us. So let's suppose that we have some random function F of X and F of X is bounded at the values of A and B notice that at the value A F of X is less than zero and at the value of B F of X is greater than zero. What the intermediate value theorem is stating is that if the values of F of X at each of the boundaries have oper signs. So for example, F of A is negative and F of B is positive, then that means the graph of F of X must pass through the X axis at least once. Meaning that there exists at least one real root for the graph F of X. With this in mind. Let's go ahead and take a look at the given polynomial. The polynomial given to us is F of X is equal to 12 X cubed minus 19 X. Plus tw 27 plus 17. And we want to determine if this has a real root on the boundaries from negative 5 to 0. So in order to start this process, what we will need to do is we will need to plug in the values of the boundaries into our function F of X. And if we see that the outputs have opposite signs, then that proves that there exists a real root by the intermediate value theorem. To start this process, let's take the value of negative five and plug it into F of X that will give us F of negative five is equal to 12 multiplied by negative five cubed minus 19 multiplied by negative five plus 17. Now, what we need to do is we need to simplify negative five cubed will simplify to give us negative 125 negative 19 multiplied by negative five will give us the value of positive 95. So what we are left with is the first term which is 12 multiplied by negative 125 which will give us negative 1500 and positive 95 plus 17 will give us the value of 112. Finally negative 1500 plus 112 will give us the value of negative 1388. So the output of F of negative five is negative. That means that F of negative five will be less than zero. Now, we're going to repeat this process. But with the second boundary zero, if we plug in zero to F of X, we get F of zero is equal to 12, multiplied by zero cubed minus 19, multiplied by zero plus 17. Anything multiplied by zero will just become zero. So what that means is that the output of F of zero is going to be 17 and 17 is a positive number, meaning that F of zero is greater than zero. So notice that we have opposite values at each end of the boundary. What this means is that by the intermediate value theorem, there exists at least one real root for the function F of X. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to use the intermediate value theorem. And I'll go ahead and see you all in the next video.

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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^3−10x+9; between -3 and -2