In Exercises 9–16, a) List all possible rational zeros. b) Use sy...

Question

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x^3+4x^2−3x−6

Accepted Answer

list the possible rational zeros of the following function. Using the rational zero theorem. Then you straight division to find an actual zero and find the remaining zeros. So the rational series theorem states that we have plus or minus are factors of P divided by our factors of Q. This ratio gives us our possible zeros. Where P. Is the constant in our polynomial and Q. Is the leading coefficient. When the leading coefficient is just the coefficient on the highest degree of the polynomial. So You can see them that pr constant is just 10. Our leading coefficient would be the coefficient on the highest degree next the third. That is just what So our Apostle restaurant ceos are given by the factors of P over the factors of cute. So factors of p. r. of 10 would be one in 10 and two insides. Thank you. There's just one and one we can rewrite this. Now supposed to minus say one is our first one. Whatever one. Samantha to everyone. Plus 2 -5/1 Plus or -10/1. Which gives us plus or minus one plus or minus two plus or minus five and plus or minus 10. So these are possible rational zeros. Now to find the actual zero, we're going to use synthetic division. So let's choose one of our zeros. I'm gonna choose negative one to start with. So let's say we have x equals negative one. Now we want to set up our synthetic division. So this is the Activision paul and ringo. We're dividing by that goes inside the box which the coefficients goes in a different place. We got X to the 3rd but she's just one in the front minus six X squared. So negative six. Then we have plus three X. So three and plus 10. Now R. zero. We're testing negative one goes out here now. Straight division. We first want to take our leading number R. one and just bring it down. Let's get one. The next part we're gonna multiply a result by are negative one negative one times one is negative one. We put it right under here then we'll just add those two numbers in that column negative six Plus -1 gives us -7. Now I did the same thing again. Now you have one times negative seven is just seven. Now we will get 3-plus 7 which is equal to 10. Finally, we'll do it again. Neither one time since the age of 10, 10 -10 gives us zero. Now this last box is our remainder. If this remainder was not zero then this wouldn't have been an actual zero. But since our maker is zero, this is one of our zeros. One, zeros is going to be X equals negative one. And so now we can write our new polynomial instant type division. We had X the third here X to the second X to the first and then sir, constant in the new polynomial. We're going one degree down. So we have X to the second. That's the first in a constant. And I remain. Our new polynomial then turns into x squared minus seven. X plus 10. Now we can use this new polynomial to determine what our other two zeros are. So we have x squared minus seven. X plus 10. We can just factor this out some exhale. We have X and x. 252 factors that will make up 10. Well as we said before, the facts of 10 are one and 10 and two and five. Let's try two and 5. Now the one that made these people make seven whenever we want to re foil them let's say we want X -2 & X -5. I've noticed this because our our last number is positive 10. We have to have two numbers multiply the equal positive 10. We're gonna have two positive numbers or two negative numbers. Since this number in the middle here is a negative seven. I'm gonna say they're both negative which means that we will get this equaling zero or x minus two equals zero. X minus five equals zero. Where we get x equals two and x equals five. I would then say our other zeros are just too. In fact with those as our zeros. The answer to this problem turns out to be answer a Okay I hope to help you solve the problem. Thank you for watching. Goodbye

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x^3+4x^2−3x−6

Key Concepts

Rational Root Theorem

Synthetic Division

Finding Remaining Zeros