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

Question

ƒ (x) = - 6 x^{3} - 25 x^{2} - 3 x + 4; k = - 4

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 one is a zero, the function given is F of X is equal to negative 20 X cubed minus 18 X squared plus six, X plus four. We're given four answer choices, options A through D and all of them show factored polynomials. OK. What's different about them is just the factors that they use. Now, we're gonna start with the information we were given. So we were told that negative one is a zero. Now, if negative one is a zero, then plus one is a factor of our function F of X. Now, if that doesn't make sense, think about this the opposite way if we have a polynomial that's factored and one of the factors is X plus one you want to solve for the zero. So you set the function equal to zero. OK. And you can look at the factors individually. Now when you look at the factor X plus one, you set that equal to zero and you get that X is equal to negative one and so negative one. Must be a zero. OK. So here we're just doing the opposite, we were given zero and we're writing out the factor. So we know that our function F of X then can be written as this factor X plus one multiplied by some function G of X. Now, we don't know what G of X is. That's whatever is left over after we factor out X plus one. But we can find G FX G of X is gonna be equal to our function F of X divided by X plus one. So let's use a long division to do this division of these polynomials. And see if we can find G of X. Once we get GEO X, we can factor it and we'll have the complete factored form of F. So starting our long division, we draw our division line, OK. To the left of our line, we have X plus one factor we're dividing by underneath it. We have F FX F FX is 20 or sorry, negative 20 X cubed minus 18 X squared plus six X plus four. Now, in order to do, do the division, we start with the left term, we have negative 20 X Q. OK. The first term of our function F of X, we divide it by the first term of our factor on the left X negative 20 X cubed divided by X gives us negative 20 X squared. We write that up above. Now we take that negative 20 X squared, we multiply it by X plus one. OK. That gives us negative 20 X cubed minus 20 X squared. Now we write that down below underneath our original function F of X OK. So that there, it's lined up in columns corresponding to the exponents. OK. So all of the X cube terms are lined up in a column. All of the X squared terms are lined up in a column and so on. Now we're gonna take the two rows we have so far and subtract we have negative 20 X cubed minus 20 execute. That gives us zero, negative 18 X squared minus negative 20 X squared gives us positive two X squared. The remaining terms we're gonna bring down, there's nothing to subtract there. And so they stay the same. We have plus six X plus four. Now we have this new function two X squared plus six X plus four. We're gonna repeat the process. We take the first term divided by the first term on the left two X squared divided by X gives us two X. We write it above, we take that two X. We just found we multiply it by the entire factor X plus one on the left hand side that gives us two X squared plus two X. We write it down below. We keep the corresponding exponents in the same columns and we subtract two X squared minus two X squared gives us zero six X minus two, X gives us four X in our constant of four, we bring down again because there's nothing to subtract. We have four, X plus four, we repeat the process again. Four, X divided by X gives us four, multiplying that four by X plus one gives us four X plus four. We write it down below hey, in the corresponding columns and we subtract those functions and you'll notice these functions are the same. We have four, X plus four minus four, X plus four, which gives us zero. OK. So we have that remainder of zero, which is what we want. Now, this function at the top that we've been writing out piece by piece is the function G of X that we were looking for the result of our division and that is negative X squared plus two X plus four. So now we can write our function F of X as X plus one. That original factor we were given multiplied by negative 20 X squared plus two X plus four. Now, one thing we like to do, we like to have a positive leading coefficient before we factor this um uh quadratic term. The second term here. So we're gonna factor out that negative. You'll notice that there's also a factor of two in each of these terms. So we can actually factor out a negative two. So when we do that, we get that F of X is equal to negative two multiplied by X plus one, multiplied by 10 X squared minus X minus two. Now, we're gonna break up this function in order to factor it, OK, you can factor it in any way that you choose. So we have F of X is equal to negative two, multiplied by X plus one. We're leaving that alone and then we're gonna write 10 X squared minus five X plus four, X minus two. Yeah. So these expressions are the same 10 X squared minus X minus two and 10 X squared minus five, X plus four, X minus two. We've just broken up that middle term. Now, when we do that, we can factor two terms at a time. Uh So we have negative two, multiplied by X plus one. Again, we're leaving that alone. Now we're gonna factor the first two terms in our second expression here. We have 10 X squared minus five X. We can factor five X from both of those. And what we have left over is two X minus one. We can do the same for the next two terms. OK? We're at it, we have four X minus two, we can factor out at two. And when we do that, we're left with two X minus one. And the way we chosen these middle terms is so that we end up with the same factor in both of these terms. OK? So now we have two X minus one in both terms. So we can factor that out. OK, we have negative two X plus one multiplied by two, X minus one. Now, when we factor out that two X minus one, what's left over is five X plus two. And so the final factored version of F of X is equal to negative two, multiplied by X plus one, multiplied by two, X minus one, multiplied by five X plus two. If we go to our answer choices and compare this to what we found, we can see that the correct answer is option A and again, that's negative two multiplied by X plus one, multiplied by two, X minus one, multiplied by five X plus two. 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. $ƒ (x) = - 6 x^{3} - 25 x^{2} - 3 x + 4; k = - 4$

Key Concepts

Polynomial Zeros and the Factor Theorem

Polynomial Division (Synthetic or Long Division)

Factoring Polynomials into Linear Factors

Factor ƒ(x) into linear factors given that k is a zero. ƒ(x)=−6x3−25x2−3x+4; k=−4ƒ(x)=-6x^3-25x^2-3x+4;\text{ }k=-4

Key Concepts

Polynomial Zeros and the Factor Theorem

Polynomial Division (Synthetic or Long Division)

Factoring Polynomials into Linear Factors

Factor ƒ(x) into linear factors given that k is a zero. $ƒ (x) = - 6 x^{3} - 25 x^{2} - 3 x + 4; k = - 4$