Show that each polynomial function has a real zero as described i...

Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2

between 2 and 3

Accepted Answer

Hello. Today we're gonna be proving that there is a real zero between one and two for the following polynomial function. So what exactly does it mean to prove that there is a real zero between these values? Well, what we're paying attention to is the values of one and two on the X axis. And if there is a sign change between one and two, then that means that the graph is going to cross the X axis at least once, meaning that there has to be at least one real zero. So how do we go about showing that there is a real zero? Well, the first thing we can do is plug in the values of one and two into our given polynomial function and see whether the outputs give us a sign change or not. So let's go ahead and start by plugging in one into our function. Plugging in one is going to give us four times one cubed minus 21 times one squared plus times one plus five. Now one raced to any power is just going to give us one. And what we are left with is four times one which is four minus 21 times one, which is negative 21 plus 16 times one, which is 16, then plus five and adding these values together is going to give us an output of four. So we know that F of one is going to give us a positive value of four. Now let's find the value for when we plug in two into our function. So if we plug in into, into the function, we get four times two cubed minus times two squared plus 16 times two plus five, two cubed is going to give us the value of eight. So we have four times eight and two squared is going to give us four. So we have negative 21 times four, 16 times two is going to give us 32. Then we are left with plus five. Now four times eight is going to give us 32 negative 21 times four is going to give us the value of 84 negative 84. Then we're adding 32 then adding five. And if we add these four numbers together, what we are left with is negative 15. So we know that F of two is equal to negative 15. Now notice that when we plug in one, then we plug in two, the output becomes a positive number then moves into a negative number. And so graphically what this means is that going from the value of one to the value of two on the X axis, the graph started off as a positive value then switched to the negative value, which means that it crossed the X axis at least once. And that means that there is a real zero that exists between the values of one and two. 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 approach this problem. And I'll go ahead and see you all in the next video.

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Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 between 2 and 3