In Exercises 39–52, find all zeros of the polynomial function or ...

Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 4x^4−x^3+5x^2−2x−6=0

Accepted Answer

Hello today we're going to be using to start this rule of signs and the rational 00. To find the zeros of the given polynomial. So the polynomial is two X to the fourth minus five X cubed minus 15 X squared Plus 10 x plus eight. So this is our original function F of X. So using this cartas rule of zeros or signs, we need to go ahead and identify all the sign changes from our original function. Well our first coefficient is positive, then the next term is negative. So that is considered a sign change. So that's one sign change. Then from negative five X cube to negative 15 X squared. The sign doesn't change. It stays negative. Then from negative 15 X squared to positive 10 X. The sign does change because we go from negative to positive. So that's a second sign change. And then finally we go from positive 10 X to positive eight. And that is not a sign change. So for F of X we have to sign changes in total and that means That there's going to be two or 0 positive zeros for the given function. Now we need to go ahead and check the possible zeros for F of negative X and F of negative X. We're gonna go ahead and plug in negative X for every X in our original polynomial. So we get two times negative X to the fourth power minus five times negative X cubed minus five times negative X squared plus 10 times negative X plus eight. Now we need to go ahead and simplify this. Well negative X to the fourth is going to give us positive X. Because negative X is raised to a positive exponent. So we have two X to the fourth. Now negative X cubed is going to produce negative X. Because this is negative X times negative X times negative X. Which is going to produce a negative value. So we have negative five times negative X cubed and that's going to give us positive five X cubed. Now we have negative X squared negative X times negative X gives us a positive value. So this is going to be minus five X squared and times negative X is going to give us negative 10 X. And then we have plus eight. So now we need to go ahead and identify all the sign changes in f of negative X. So we have negative two X to the fourth to positive five X cubed. Those are both positive. So there's no sign change. Then from positive five X cubed to negative five X squared. The number goes from positive to negative. So we have a sign change here then from negative five X squared to negative 10 X. There is no sign change and from negative 10 X to positive eight, we have a sign change. So there are two sign changes for F of negative X. And that means that there's going to be two or negative zeros for a given polynomial. So now we need to go ahead and actually factor our polynomial in order to find the zeros. Well, we're going to use the PQ method where P is the factors of our last term and Q is the factors of our first term. So P over Q is equal to plus or minus the factors of eight over the factors of two. And the factors of eight is going to be plus or minus +124 and eight. And the factors of two is just going to be one or two. So we need to divide all of our possible factors of eight by one. Then all the possible factors of eight by two. Once we do that and not including any of the repeating factors. The possible zeros are going to be poss er minus 124, 8.5. So we need to go ahead and pick one of these factors to synthetically divide a polynomial. By that way we can be able to figure out what our first zero is. Well, let's go ahead and pick negative two. And we're going to divide the coefficients of our original polynomial by -2. So the coefficients of our original polynomial is two negative five negative 15. 10 and eight. Now if negative two is an actual zero, then it's going to produce a zero as a remainder for the synthetic division. So let's go ahead and proceed with the synthetic division. We start by bringing down to then two times negative two is going to give us negative four and we put that underneath the next coefficient. And we're going to repeat this process until we reach the very last coefficient. So negative five minus four is going to give us negative nine, negative nine times negative two is going to give us positive 18 and negative 15 plus 18 is going to give me +33 times negative two is going to give me negative six and 10 minus six is going to give me four and four times negative two is going to give me negative eight and eight minus eight is going to give me zero. So since negative two produced a remainder of zero for the synthetic division negative two is a zero of our original function. So X is equal to negative two. That is the first zero of our function. And now we need to list out our new factored polynomial but with one power less than the original. So using our new coefficients, we have two X cubed minus nine, X squared plus three X plus four. And we need to go ahead and factor this again. What we're going to repeat the process of the PQ. So the PQ is going to be 4/2 and the possible factors is going to be plus or minus 1 - 4/1 and two. And simplifying this is going to give me 1, 2 and 1/2. So let's go ahead and try negative half as another possible zero. So negative half is going to synthetically divide the coefficients of our new polynomial and that's going to be too negative 93 and four. And now we're going to continue this process and hopefully get a zero as the remainder. So we bring down the two and then we want to multiply two by negative half which is going to give me negative one and negative nine minus one is going to give me negative 10 negative 10 times negative half is going to give me positive five and five plus three is going to give me eight. And finally eight times negative half is going to give me -4 and 4 -4 is going to give me zero. Now since negative have produced a zero as the remainder for the synthetic division X is equal to negative half is another zero of our polynomial. And now we want to go ahead and let's start a polynomial with the new coefficients. So we have two, X squared minus 10. X plus eight is equal to zero. And we can go ahead and simplifying this by dividing everything by two. And doing so is going to give me X squared minus five X plus eight. I'm sorry not plus eight plus four is equal to zero. And this is a fact Herbal Trin Amiel because this try no meal can factor two, X minus one times X minus four. And using our zero property, we can set both of these factors equal to zero. So we have x minus one is equal to zero and x minus four is equal to zero. I'm gonna go ahead and add one to both sides of this equation and X is going to equal to one and that is another zero for a polynomial. And finally I'm gonna add four to both sides of this equation and I get X is equal to four and that's gonna be the other zero for a polynomial. So just listing out all the zeros that we've obtained, we have X is equal to negative two, negative half one and four. And this is gonna be the total amount of zeros for our original polynomial. And with that being said, the answer to this problem is going to be C. So I hope this video helps you understand how to find the zeros of a higher order polynomial and how to use discard this rule of signs and I'll go ahead and see you all in the next video

Key Concepts

Rational Zero Theorem

Descartes's Rule of Signs

Graphing Polynomial Functions