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 -1 and 0

Accepted Answer

Hello. Today we're going to prove that there is a real zero between negative one and zero for the following polynomial function. So the question is what does it mean to prove that a real zero lies between the values of negative one and zero? While this is referring to the values of negative one and zero on the X axis. And what this is trying to show is that if we plug in the values of negative one and zero into the function, and we show that there is a sign change between the two values. That means that the function would have to cross the X axis at least at one point when going from negative 1 to 0. So we can go ahead and start this process by plugging in negative one and zero into the polynomial function. Let's go ahead and start by plugging in negative one. If we plug in negative one, we get four times negative one cubed minus times negative one squared plus 16 times negative one plus five. Now negative one raced to an odd power is going to give us negative one and four times negative one is going to give us negative four. Now negative one raced to an even power is going to give us positive one and negative 21 times positive one is going to give us negative 21 16 times negative one is going to give us negative 16. Then we are adding five at the end and negative four minus 21 minus 16 plus five is going to give us the value of negative 36. So we know that when plugging a negative one to the function, we get the value of negative 36. Now let's plug in the value of zero. If we plug in the value of zero into our function, we get four times zero cubed minus 21 times zero squared plus 16 times zero plus five. Now zero, raced to any power is going to give us zero and the times zero is going to give us zero. That means that the first three terms in our polynomial function are going to cancel out and that is going to leave us the value of positive five. So we know that when plugging in zero into the function, we get the value of positive five. Now notice that when we plug in negative one, we get a negative value, then when we plug in zero, we get a positive value. This shows that there is a sign change from going negative 1 to 0. And that means that the graph switches from a negative value to a positive value when traveling between negative one and zero, which means that there has to be at least one real zero between negative one and zero. And with that being said, the answer to this problem is going to be B 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 -1 and 0