For each polynomial function, find all zeros and their multiplici...

Question

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=3x(x-2)(x+3)(x^2-1)

Accepted Answer

Hey, everyone in this problem, we're asked to identify the zeros and their multiplicities. For the following function F FX is equal to seven X multiplied by X minus nine, multiplied by X plus four, multiplied by X squared minus 25. Option A negative five, negative 405 and nine. All of multiplicity one. Option B negative nine, negative 504 and nine. All of multiplicity one option C zero of multiplicity seven, all of the remaining zeros in option C are the same as option A and then option D, we have zero with a multiplicity of seven and then we have negative 49 and 25 all with multiplicity of one. Now we're gonna start by rewriting this function F of X. And the function we've been given is mostly factored already which helps a lot. So we have seven X multiplied by X minus nine, multiplied by X plus four multiplied by X squared minus 25. Now, these are all linear factors except for the last one X squared minus 25. So we wanna go ahead and factor that further. If we can, when we look at this, we want to recognize it as a difference of squares. I recall that if we have a squared minus B squared, you can write this as A minus B multiplied by A plus B. Now, in this case, A is going to be X, OK. This first term is X correct and B is going to be five because B squared is equal to 25. So we can write our function F of X as seven X multiplied by X minus nine, multiplied by X plus four, multiplied by X minus five, multiplied by X. OK. We use that difference of squares to write X squared minus as X minus five multiplied by X plus five. Now, in order to find the zeros, we want to set this function equal to zero and why to be equal to zero. So we have zero is equal to seven X multiplied by X minus nine, multiplied by X plus four, multiplied by X minus five, multiplied by X plus five. Now, all of these terms on the right hand side are factored and so they're multiplied together. So if we want the left hand side to equal zero, as long as one of these factors on the right hand side is equal to zero, we're gonna satisfy that equation. OK. So setting any of these factors on the right hand side equal to zero is going to give us a solution. So if we start with our seven X term seven X equals zero tells us that X must equal zero. Moving to the next term we have X minus nine. If we set this equal to zero, we get that X is equal to nine. Our next factor X plus four, we set this equal to zero and we get that X is equal to negative four. OK? We have to subtract it to move it to the other side. We have our factor X minus five setting this equal to zero. We need to add five to both sides to isolate X. We get that X is equal to five. And finally, our last factor X plus five, we set this equal to zero and we get that X is equal to negative five. And so we found five zeros, we found X equals zero, X equals nine, X equals negative four, X equals five and X equals negative five. Now we need to figure out the multiplicities and the multiplicity of A zero is the number of times it shows up in a polynomial. OK. So the number of times a factor associated with a zero shows up in our function. Now, when we look at our function F FX, all of these zeros only occur once and we don't have any squared or cube terms on any of these factors. And so they all have a multiplicity of just one. So we have our five zeros negative five, negative 405 and nine all with the multiplicity of one which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=3x(x-2)(x+3)(x^2-1)

Key Concepts

Polynomial Functions

Zeros of a Polynomial

Multiplicity of Zeros