For each polynomial function, one zero is given. Find all other z...

Question

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=-x^4-5x^2-4; -i

Accepted Answer

Hey, everyone in this problem, we're asked if three, I is a zero of the following polynomial function solve for all the other zeros. The function we're given is F of X equals X to the exponent four plus 25 X squared plus 144. We're given four answer choices. All four of them include X equals negative three. I. Option A also has X equals four and X equals negative four. Option B also has X equals five and X equals negative five. Option C also has X equals four, I and X equals negative four, I and option D also has X equals five, I and X equals negative five. I. Now, if we have X equals three, I is a zero, then we automatically have X equals negative three, I as a zero as well. OK? Our co complex roots always come in comp um complex conjugate pairs. OK? They always come in pairs. And so if three, I is a root then negative three, I is a root. All right. So we've already found one of our additional roots, X equals negative three, I. Now if X equals three, I is a zero, then X minus three, I is a factor. And if X equals negative three, I is a zero, then X plus three I is a factor. OK? So we have these two factors. That means that we can write our function F of X as these two factors, X minus three, I multiplied by X plus three. I multiplied by some function G of X. OK. And G of X is gonna be what's left over when we remove those two factors. Now let's simplify X minus three, I multiplied by X plus three. I, we get that F of X is equal to X squared minus three I X plus three I X minus nine I squared. OK. Multiplied by G X. Now this negative three I X plus three I X that's gonna add to zero. And we have negative nine I squared. We call that I squared is defined as negative one and so negative nine I squared, negative nine multiplied by negative one, which gives us positive nine. We have F FX is equal to an X squared plus nine multiplied by G of X. Now, in order to find G of X, we need to divide by X squared plus nine. OK. So to find G FX that function that's left over, that's gonna contain those other zeroes we're looking for, we're gonna divide F of X I, this factor X squared plus nine that we have. But we're gonna do that using long division, we go down, give ourselves some more room and we're going to set up our long division. OK? And we're gonna write out the function we have F of X is X to the exponent four. And we're gonna include the X cube term, OK. In our function, we don't have an X cube term. Um We wanna include it when we're doing our long division and we can just write it as zero X cubed plus 25 X squared plus 144. And so that goes underneath our line in our long division. And on the left hand side, we have what we're dividing by which is X squared plus nine. Now, the first thing we're gonna do, we're gonna take the first value N R function X to the exponent four. And we're gonna divide it by the first term on the left hand side, X word X to the exponent four divided by X squared is going to give us squirt. We're gonna write that above. Now, we're gonna take that X squared and multiply it to this entire term. On the left hand side, X squared multiplied by X squared plus nine. This gives us X to the exponent four zero X Q plus nine X squared. We write this underneath our original function. We're gonna draw a horizontal line and we're gonna subtract our original function minus this new function that we've found. We have X to the exponent four minus X to the exponent four, which is going to give us zero zero X cubed minus zero X cubed gives a zero 25 X squared minus nine X squared gives us 16 X squared. And we bring this 144 down because there's no constant term in the new function we've found yet. So now we have 16 X squared plus 144 and we're gonna go through the same process again, we're gonna take the first term there 16 X squared and divide it by the first term on the left hand side, X squared, 16 X squared divided by X squared gives us 16. We write it up above. So now above, we have X squared plus 16, we take that new value of 16 we found and we multiply it by the left hand side. So we have 16, multiplied by X squared plus nine that gives us 16 X squared plus 144. We write that down below underneath the other function we had. And then we subtract. Now these two equations are the exact same. So when we subtract them, we are going to get zero, we have a remainder of zero, which is exactly what we want in order for this function to factor nicely. So we found that when we divide G um f of X by X squared plus nine, what we're left with is the function above our division line which is X squared plus 16. So now we can write that F of X is equal to X squared plus nine multiplied by X squared plus 16. Now, in order to factor X squared plus 16, it's gonna be very similar to factoring X squared plus nine. So we know that the factors for X squared plus nine are X minus three, I and X plus three, I and similarly, for X squared plus 16, our factors are going to be X minus four, I and X plus four. I. Now you can imagine if we multiply these two, we get X squared, then we have plus four I X minus four I X, those two terms will divide out and then we get negative 16, I squared I squared is equal to negative one. And so that gives us positive 16. So we get X squared plus 16, which is exactly what we had. OK. So we've completely factored this function F FX is equal to X minus three. I multiplied by X plus three. I multiplied by X minus four. I multiplied by X plus four. I, we wanna set this equals to zero to find the zeros. OK. So we set the left hand side equal to zero. The right hand side stays the same. Now we have all of these terms multiplied together equal to zero. If any one of these terms is equal to zero, the entire right hand side will equal zero because they're being multiplied from the first term. If we set this equal to zero, we get X is equal to three I, which is one of the roots we already had, we knew this. So this is just a good check to make sure that we've written this function properly. Same with the next one. We have X plus three, I is equal to zero, which tells us that X is equal to negative three I. So that's good. We expected that one. Our next term X minus four, I, we set that equal to zero, which tells us that X is equal to four, I in our final factor X plus four, I, we set that equal to zero and that tells us that X is equal to negative four I and so the four zeros we have are X equals three I X equals negative three I X equals four. I and X equals negative four. I. Now we were already given the first factor of three I. And the problem, so the problem is just asking what are the three other factors or the three other zeroes? Sorry. And we found that they are X equals negative three I X equals four, I and X equals negative four I which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=-x^4-5x^2-4; -i

Key Concepts

Polynomial Functions

Zeros of a Polynomial

Complex Conjugate Root Theorem