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^2−2x+1; between -3 and -2

Accepted Answer

Welcome back. I'm so glad you're here. We are given the polynomial function F of X equals to X cubed plus four X squared minus six X plus three. We are told that there is a real zero between negative four and negative three. And we need to prove it. Using the intermediate value theorem. Well, we recall from previous lessons that the intermediate value theorem tells us that if F of negative four and F of negative three have opposite signs, then at least one real zero lies between negative four and negative three. So let's start off by testing that and finding out what F of negative four is equal to. We will substitute a negative four anywhere. There was an X in our original F of X equation. So now two times negative four, cute plus four times negative four squared minus six times negative four plus three. Let's start off by working our exponents. I'll bring down this two times and now negative for cute. That is negative 64 plus four times negative four squared is a positive 16 minus six times negative four plus three. Now let's do our multiplication two times negative 64 is a negative 128 plus four times 16 is plus 64 minus six Times negative four is going to be a plus 24. Bring down our plus three. Now doing all of that addition there will give us a negative 37. So F of negative four is negative 37. And what's important to note here is that this is a negative value. Now we are going to test out f of negative three to see if that gives us a positive value. If there's a sign change in between f of negative four and F of negative three. Alright, so we're going to plug a negative three. And everywhere there was an X in our original F of X equation. So two times negative three cubed plus four times negative three squared minus six times negative three plus three will start off with our exponents bring down that two times negative three cubed is negative 27 plus four times negative three squared is going to be a positive nine minus six times negative three plus three. Now let's do the multiplication, two times negative 27 is negative 54 plus four times nine is plus 36 minus six. Times negative three is going to be plus 18 plus three. Doing all of that addition. There is going to give us a positive three. So f of negative three equals positive three. And yes, that is a sign change. We went from a negative number to a positive number. So our intermediate value theorem tells us yes. Yes there is a real zero in between negative four and negative three. We look at our answer choices and this matches with answer choice. D Well done. We'll catch you 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^2−2x+1; between -3 and -2

Key Concepts

Intermediate Value Theorem

Polynomials

Sign Change

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