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. f(x)=x^4−2x^3+x^2+12x+8

Accepted Answer

Hello. Today we're going to be using discard this rule of signs and rational zero theorem to find all the zeroes of the given function. So the function is F of X is equal to X cubed minus X squared minus 14 X plus 24. So for discard this rule of signs, we need to go ahead and identify all the sign changes in F of X. So the leading coefficient is positive, then we go to a negative coefficient. So there's one sign change for the first two terms. Then to the 2nd and 3rd term we have negative two negative. So there's no sign change then to the third or fourth term. We have negative to positive. So there's one sign change here as well. So for F of X there's a total of two sign changes And that means that there's going to be two or 0 positive zeros for our given function. Now we need to go ahead and check the sign changes for F of negative X. Well, what we're going to do is plug in negative X for all the given X is in our function. So we have negative X cubed minus negative X squared minus 14, negative X plus 24. Now negative X is raised to an odd exponent in the first coefficient. So that's going to produce a negative value. So we have negative X cubed now since in this next term negative X is squared and negative X times negative X produces a positive value but there's a negative sign outside. So we have minus X squared negative 14 times negative X is going to give me positive 14 X. And then we have plus 24. Now let's go ahead and identify the sign changes here. The first term is negative, then the second term is negative. So there's no sign change. The second term is negative and the third term is positive. So there's one sign change here and then the third term is positive and the fourth term is positive. So there's no sign change from the third to fourth term. So for f of negative X, there's one sign change And that means that there's going to be one negative zero for our given function. Now we actually have to factor or find the zeros of our function. And we're going to use the PQ method, P is going to be 24 Q is going to be X cubed. So P and Q represent the possible factors of those given numbers and they have the potentiality of being positive or negative. So we have the factors of 24 over the factors of one and that's gonna be plus or minus the factors of 24 which is going to be +123468, 12 and 24. All over the factors of one, which is one. And once we simplify this, the possible zeroes for our given function is going to be 123468, 12 and 24. So we want to go ahead and pick one of these possible zeros and divide our original function by it that way through the synthetic division. If we get a remainder of zero then that's going to be a zero for the given function. Let's go ahead and pick negative four. If we synthetically divide our function by negative four and get a remainder of zero, the negative four should be a zero of that function. So let's list out the leading coefficients of our function and that's going to be one negative one, negative 14 and 24. So we start this process by bringing down one. Then we multiply one by negative four and that's going to go under the next leading coefficient one times negative four is going to give us negative four and then negative one minus four is going to give us negative five. And we're going to repeat this process all the way until the very last coefficient negative five times negative four is going to give me positive 20 and negative 14 plus 20 is going to give me +66. Times negative four gives me negative 24 24 minus gives me zero. So negative four produced a zero as the remainder for the synthetic division. So X is equal to negative four is a zero for a given function. And now we need to go ahead and list out our new factors polynomial with a bleeding power less than the original. So we have X squared minus five X plus six. And we're gonna set that equal to zero because we're trying to find the remaining zeros. Now, this is a fact arable, Trin Amiel because this factor is two, X minus three times X minus two. And using our zero property we can set both of these factors equal to zero. So we get X -3 is equal to zero and X -2 is equal to zero on the left hand side. I'm just gonna go ahead and add three to both sides of this equation and X is going to equal to three and that's our other zero for the equation. And on the right hand side I'm gonna go ahead and add two to both sides of the equation. So X is going to equal to two and that's gonna be the last zero for our function. So in total there are four zeros for our given function X is equal to negative 42 and three. And that's going to be the answer to our problem. And that means that the solution to the overall problem is going to be be So I hope this video helps you in understanding how to use discard this rule of signs and how to factor the higher order polynomial. And I'll go ahead and see you all in the next video

Key Concepts

Rational Zero Theorem

Descartes's Rule of Signs

Graphing Utility