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+x^2−4x−4

Accepted Answer

List the possible rational zeros of the following function using the rational zero theorem. Then use synthetic division to find the actual zero and find the remaining zeros using the quotient from division where our function F of X is X, the third plus three X squared minus 10, X minus 24. Now, to solve this, we need to use the rational zero standard which they set are possible. Rational zeros are given by the ratio plus or minus P divided by plus or minus Q which are the factors of P and Q. Now, in our, our function P is our constant turn, which is our 24. In this case, our Q is our leading coefficient which is on the X to the third which is the one. So P will be 24 Q will be one and these are the factors of both of these numbers. So let's first find the factors of 24 for 24. We have one and 24 two and 12, three and eight and four and six. For Q, we just have one and one. So are possible rational zeros then will be plus or minus one divided by one plus or minus two, divided by one plus or minus three, divided by one plus or minus four, divided by one plus or minus six, divided by one plus or minus eight, divided by one plus or minus 12, divided by one and plus or minus 24 divided by one. So we can rewrite these once more plus or minus one plus or minus two plus or minus three plus or minus four plus or minus six plus or minus eight plus or minus 12 and plus or minus 24. So now we will use synthetic division on our original polynomial to see if we can find a possible zero. For example, let's say we use the rational zero, negative two and we will perform state division with this. So we have NATA two as a possible zero and we have our coefficients from our function which will be 13, negative 10 and negative 24. If our remainder at the end is zero, then we confirm this is a actual zero. So when synthetic vision, we first bring down our first number one, we can then multiply our resulting one with our negative two and put it underneath our next term. So one multiplied by negative two gives us negative two. Now we add these together three plus N two is just one. We do it again. Now one multiplied by negative two gives you negative two, which added together, we get negative 12. 12, which is negative multiplied by negative two, gives us positive 24 add those together and we get zero. This means our resulting polynomial after our division is X squared plus X minus 12. With our current zero X equals negative two, which as a factor is X plus two, we cannot make use of our resulting polynomial and factor it to find X to X squared plus X minus 12. We can factor this out. We factor it. If X, if we want factors of 12, that will give us positive X. Let's say these are plus four and minus three. If we set the s equal to zero, we can solve for our X X plus four equals zero, X minus three equals zero to get X equals negative four and X equals positive three. We now have three possible zeros, right? We have three actual zeros. Now we have negative two, positive three and negative four. Those are the answer to our problem, which means the star problem turns out to be answer C OK. 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+x^2−4x−4

Key Concepts

Rational Root Theorem

Synthetic Division

Finding Remaining Zeros