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

Find the zero in part (b) to three decimal places.

Accepted Answer

Hello, today we're going to be calculating the value of the zero that lies between one and two for the following polynomial function. Now, in a previous video, we proved that there is a zero between one and two. For a given function, we plugged in the value of one into our function. We gave us the value of four and we plugged in the value of two which gave us negative 15 since there was a sign change in the values going from 1 to 2 graphically. What that meant was that looking at the values of one and two on the X axis, the graph went from a positive value to a negative value, which means that there's going to be one real zero between the values of one and two. But now the question is how do we get that value for that zero? Well, what we can go ahead and do is use the rational zero theorem and what we want to do now is list out all the possible zeros for this value that we're looking for. So we're going to use the PQ method of getting the rational zeros. We're going to list out all the possible factors of the last term, which are going to be the P factors and all the possible factors of the coefficient of the first term, which we're going to call the Q factors. Now five is a prime number, which means the only possible factors are going to be one and five. And the possible non repeating factors for four are going to be 12 and four. And now to get our list of possible rational zeros that we're gonna label our zeros as P over Q. And that is going to give us the list of one comma 5/1 comma two, comma four. And simplifying this is going to give us plus or minus +11 half, 1/4 +55 halves and five fourths. So what we just listed here is a list of all the possible zeros for the rational function. But keep in mind, we only want to find the zero that lies between one and two, not including the values of one and two. So to simplify testing all of these values, we only want the positive values that lie between one and two and the only values that satisfy that range within our possible list of values is going to be 5/2 and 5/4. And so now what the rational zero theorem says is if we plug in a value into our function and it gives us zero, then that is going to be the zero for the function. So let's go ahead and test out the value of 5/4. So we're gonna plug in the value of 5/4 into our function. And that is going to give us four times 5/4, raised to the power of three minus times 5/4, raise the power of two plus 16 times 5/4 plus five. Now, there's a bit of simplification that we can do here. 1st 5/4 race to the power of three is going to give us the value of 100 and 25/64. So we have four times 1 25/ and 5/4 raced to the power of two is going to give us 25/16. So we have negative 21 times 25/16. Then we have 16 times 5/4. We can rewrite 16 as 16/1, multiplied by 5/4. And through cross reduction, we can reduce this to give us four and one and then plus five at the end. Now we can further reduce things because in the beginning, we can rewrite four as 4/1. And by using cross reduction, we can cancel that out and reduce 64 to give us 16. And the same thing can be applied with negative 21 we can rewrite negative 21 as 21/1. And since we want to keep the debt denominator of 16. We're just gonna multiply 21/1 with 25/16. So what we are left with is 100 and 25/ minus 21/1 times 25/16, which is going to give us 525/ plus four times five, which is going to give us 20 plus five. Now we can subtract the first two fractions together to give us negative 400/16 and 416 are both divisible by 16. This is going to reduce to give us 25/1. And if we add 25 together, that's going to give us 25. And so what we are left with is negative 25 plus which is going to give us zero. So that means that when we plugged in the value of 5/4 into our function, it gave us an output of zero. And through the rational theorem, rational zero theorem, that means that 5/4 is going to be the rational zero between the values of one and two. Now keep in mind the instructions want us to round this value to two decimal places and 5/4 turned into a decimal is going to give us the value of 1.25. So the value of the zero between one and two is going to be 1.25. And with that being said, the answer to this problem is going to be c 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 Find the zero in part (b) to three decimal places.