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)=4x^3-37x^2+50x+60Find the zero in part (b) to three decimal places.

Accepted Answer

For the following polynomial function, evaluate the zero that lies between six and seven and write with four decimal places. F of X equals seven X to the third minus 1 27 X squared plus 5, 36 X minus 74. And we have four possible answers between six and seven. Now, to solve this, we once make use of the possible rational root theorem which will help us determine all the zeroes we have in our function that tells us our possible rational routes are given by a plus or minus factors of P divided by factors of Q where P is a constant time at the very end. And Q is the coefficient of highest degree. Mean R P is 74 Q is seven. Let's find the factors of both of these 74. We have 1 74 and two and 37. For seven. We just have one and seven. We can go ahead and form our possible roots. Now we will have plus or minus factors of P. So let's say we start with one, one divided by one plus or minus two, divided by one plus or minus 37 divided by one and plus or minus 74 divided by one. You can also use our seven to get plus or minus one sevens plus or minus two sevens plus or minus 37 7th and plus or minus 74 7th. Now, we can see if one of these are a zero. For example, let's choose X equals 1/ and we can plug this into our function F of 1/ equals seven multiplied by 1/7 to the third minus 27 multiplied by 1/7 squared plus 36 multiplied by 1/7 minus 74. If I were to plug into a calculator, we will see that this equals zero. That confirms this as one of our zeroes. Now that we have a zero, we can use long division to determine our other zeroes. We have X equals 1/7 which can be transformed by multiplying both sides by seven to get seven X equals one, which can be further transformed by moving the one over to the other side to get a seven X minus one equals zero. This gives us a factor we can divide by, you will have seven X minus one as our bottom of our division. And then we have our equation. So next to third minus 1 27 X squared plus 5 36 X minus 74. Let's go ahead and do long division. Now we will take our seven X to the third and divide it by seven X that will just give us X to the second that goes at the top. Now we will subtract and multiply X squared multiplied by seven X minus one. That is a seven X, the third minus X square. You will get negative 1 26 X squared remaining. After we do the division or multiplication, I should say and our subtraction, no, we have negative 1 26 X squared divided by seven X. That will give us negative 18 X that will go off the top. And now we will multiply it once again with seven X minus one minus negative 1 26 X squared plus 18 X. We can subtract and we will get 5 36 X minus 18 X which gives us 5 18 X can now do this one more time. We have 5 18 X divided by seven X which gives us 74. We will add 74 at the very end lying again, you subtract 74 multiplied by 18 X minus 74. Bring down our 74 we get zero. This means our two factors now are seven X minus one multiplied by X squared minus 18 X plus 74. Let's find the factors of X squared minus 18 X plus 74. Now to determine what R zero could be. Yep X square minus 18 X plus 74. Now this doesn't factor evenly. So we have to use quadratic formula. Quadratic formula is given by X equals negative B plus or minus the square root of B squared minus four AC all divided by two A. You would get negative negative 18 plus or minus square root negative squared minus four, multiplied by one multiplied by over two, multiplied by one. Now, we can simplify it 18 plus or minus square root. 3 24 on a 74 multiplied by four is this 2 all divided by two, which gives us 18 plus or minus square. It 3 24 minus 2 96. That would be 28 divided by two. Finally, we simplify everything 18 plus or minus two squared to seven divided by two, which gives us nine plus or minus the square root of seven. Now, we can approximate, you know, X equals nine plus A squared of seven, which approximates to 11.6458. You have also X equals nine minus squared to seven, which approximates to 6.3542. Looking for the answer that is between six and seven, which that answer is 6.3542. If you look at our possible answers, we then determine the answer to our problem is answer A OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Struggling with College Algebra?

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)=4x^3-37x^2+50x+60Find the zero in part (b) to three decimal places.

Zeros of Polynomial Functions practice set