Factor ƒ(x) into linear factors given that k is a zero. See Examp...

Question

Factor ƒ(x) into linear factors given that k is a zero. See Example 2. ƒ(x)=2x^4+x^3-9x^2-13x-5; k=-1 (multiplicity 3)

Accepted Answer

Hey, everyone in this problem, we're asked to factor the following function to express it as a product of its linear factors. If negative six with the multiplicity of three is a zero, the function we're given is F of X is equal to X to the exponent four plus 17, X cubed plus 90 X squared plus 108 X minus 216. We're given four answer choices options A through D which are all factored polynomials. Now we're told that negative six is a zero. OK? So if negative six is a zero, that tells us that X minus negative six is a factor of our polynomial. And you can think of this the opposite way. OK? If that's confusing, if you had a factor X minus negative six and you want to define the zeros, you set the function equal to zero. Yeah. So you can set the factor equal to zero. And that would give you that X is equal to negative six. OK? So you get the zero of negative six. So we're just working backwards here. We have the zero. What is the factor? Now, X minus negative six? This can be written as oh last six. But we're told that it has a multiplicity of three. Now recall that a multiplicity tells us how many times a factor appears in our function. OK. So a multiplicity of three means this factor shows up three times. And so our function F of X can be written as this factor X plus six to the exponent three multiplied by some function G of X. Now we're writing into the exponent three because of that multiplicity of three. Now, this factor or this function G of X is whatever's left over when we factor out our X plus six to the exponent three, in order to find G of X and we need to find G FX in order to figure out the rest of the factors that we have in our function. In order to do that, we can divide F of X by X plus six cubed. So we're gonna divide of by X plus six Q in order to get G O X. Now, when we're dividing polynomials, we don't wanna have them factored, we wanna have them expanded in that standard form. So we need to expand X plus six cubed, X plus six cubed. While this is equal to X plus six squared, multiplied by X plus six X plus six squared. You can expand any way you'd like whether you use the foil method or some other method. We're gonna get X squared plus 12 X plus 36. This is multiplied by X plus six and we're going to distribute these terms. So every term in our first factor needs to be multiplied to every term in our second factor. So we're gonna have X squared multiplied by X gives us X cubed X squared multiplied by six. Gives us six X squared. We move to the next term 12 X multiplied by X gives us 12 X squared, 12 X multiplied by six, gives us 72 X. Next term 36 multiplied by X gives us X and 36 multiplied by six, gives us 216. OK. So now we've multiplied every term in our left factor to every term in our right factor. And we can simplify this. So X cube, we only have a single execute. So we get X cubed, we have plus six X squared plus 12 X squared. This gives us plus a squirt plus 72 X plus 36 X gives us plus 108 X. And then finally, we have our constant term of plus 260. OK. So now we're taking our function F of X is equal to X to the exponent four plus 17 X cubed plus 90 X squared plus 108 X minus 216. And we are dividing it by X plus six cubed, which we expanded and found is equal to X cubed plus 18 X squared plus 108 X plus 216. OK. There's a lot of terms, a lot of coefficients and exponents. So it can be easy to mix up your numbers here. So take your time, double check your work and just make sure that you're getting all of these factors correctly. Now, in order to do this division, we're gonna use long division, we're gonna draw out a long division line and underneath it, we're gonna write our function F of X X to the exponent four plus 17 X cubed plus 90 X squared plus 108 X minus 260. On the left hand side, we're gonna write out the function we're dividing by which is X plus six or QED X Q plus 18 X squared plus 108 X plus 216. Let me write that 216 just down below. So it's not so squished. OK. Now, in order to do the division, we take the first term under our line which is X to the exponent four, we divide it by the first term on the left hand side, X Q X to the exponent four divided by X cube gives us X. We write that X up above and now we take that value X that we found and multiply it by the entire term on the left hand side by what we're dividing by. OK. So we have X multiplied by X cubed gives us X to the exponent four, X multiplied by 18 X squared gives us X cubed X multiplied by X gives us 108 X oops, sorry, 108 X squared, an X multiplied by 216 gives us 216 X. Now we write these underneath our original function but we keep the columns consistent. OK. So the X to the exponent four terms are in a column, the X cube terms stay in a column, the X square terms, et cetera. And now we're gonna subtract in those call ups. We have X to the exponent four minus X to the exponent four that gives us zero 17 X cubed minus X cube gives us negative X cubed 90 X squared minus 108 X squared gives us minus 18 X squared, 108 X minus 216 X minus 108 X in our resulting function. And finally, this minus 216 we can just bring it down because we have nothing to subtract. And now we have this new function underneath negative X cubed minus 18 X squared minus 100 and eight X minus 216. And we're gonna do the same process. Again, we take the first term negative X cubed, we divide it by the first term on the left X cubed. This gives us a negative one. We take that value of negative one. We write it above and then we multiply it by the entire left hand side. So negative one multiplied by X cubed plus 18 X squared plus 108 X plus 216. This gives us negative X cubed minus 18 X squared minus 108 X minus 216. We write it down below underneath that function we had. And you'll notice that these are the exact same. So when we subtract, we're just gonna get zero and we have a function subtracted by the exact same function we get in zero. So our remainder is zero, which is great. The function of the top that we wrote down that is X minus one. That's gonna be the function G of X and that's the result of our division. So that's the function G of X that we were looking for that finishes off the factored form of our function. And so our function F of X is going to be equal to X plus six cubed. The original factors we had based off the zeros, we were given multiplied by this extra factor X minus one that we found through long division. That is what we were looking for. We were looking for the factored form of F FX. We go up and compare our answer choices and remember that these factors are multiplied together so we can change the order that the factors appear in, still have the same answer. And so the correct answer here is option C X minus one multiplied by X plus six cubed. Thanks everyone for watching. I hope this video helped see you in the next one.

Factor ƒ(x) into linear factors given that k is a zero. See Example 2. ƒ(x)=2x^4+x^3-9x^2-13x-5; k=-1 (multiplicity 3)

Key Concepts

Factoring Polynomials

Zeros and Multiplicity

Synthetic Division