Use synthetic division to determine whether the given number k is...

Question

Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). 
ƒ(x) = 2x^3 - 6x^2 -9x + 4; k=1

Accepted Answer

Hey, everyone in this problem for the given polynomial function, we're asked to perform synthetic division to test if K equals four is a zero. In case it's not, we're told just to evaluate F F K. Now, the function we're given is F of X is equal to three X cubed minus 18, X squared minus 25 X plus 28. We're given four answer choices. Option A yes, option B no, a negative 352. Option C no and negative and option D no, negative 128. Now, if we want to test, if K equals four is zero and then we wanna look at the factor X minus four. OK? If K equals four as a zero, then X minus four is going to be a factor of this polynomial. So we're gonna take our polynomial F of X which is three X cubed minus 18, X squared minus 25 X plus 28 divided by X minus four. All right. So four is a zero, then X minus four is a factor of this polynomial. And what that means is when we take this polynomial F of X and divided by X minus four. We're gonna get a nice quotient out of this with no remainder. OK. It's gonna divide nicely into another polynomial. All right. So let's do our synthetic division. We're gonna set up our table and we're gonna start with the number on the left hand side of the table. And that's gonna be the root of the denominator. In this case, the root of the denominator is four. OK? That's the route we're trying to test. Then we're gonna fill out the top row of our table using the coefficients from our function F FX in the numerator. So we're gonna start on the left-hand side with the coefficient corresponding to the highest exponent term. Our highest exponent term is execute and the coefficient there is the so three is the first number in that first row of our column or of our table. Then we're gonna move to the next column over staying in that first row. And we're just gonna look at the coefficient of the next highest exponent term. OK. It's X squared coefficient negative 18. So we write negative 18, we move to the right now we're on our X term coefficient negative 25. So negative 25 goes in our table. And finally, we have our constant term of 28 that is the last column of that first row of our table. So now we have our synthetic division table set up, we wanna actually perform the synthetic division. So we're gonna start on the left hand side, we're gonna take the three and we're gonna bring it down and underneath our table. Now, when we have a value underneath our table, we're gonna multiply it to the value outside our table on the left hand side. So we have three multiplied by four, which gives us 12. We're gonna write that in the next column over and in the second row of the table. So now we have this complete column we're gonna add. So we have negative 18 plus 12 which gives us negative six. We write that down below underneath our table in this second column that we're working in. Now we have a new value underneath our table negative six. We're gonna multiply that to the value on the left hand side, negative six multiplied by four gives us negative 24. We write that in the next column over in that second row of the table. And now we have another complete column. So we're gonna act negative 25 plus negative 24 gives us negative 49. We write it below. In that same column, we have a new number below our table negative 49. We multiply to the number on the left four. This is gonna give us negative 196. We write it in the next column over in that second row. Now we have a complete column again. And we're going to add, we have 28 plus negative 196. This gives us negative 168. Now, our synthetic division is done and we just need to interpret the results. I'm gonna draw a dashed line separating the last column on the right hand side of our synthetic division. OK? And this bottom number, the far right bottom number is the remainder of this division. OK? So in this case, the remainder is negative 168. Now remember what I said, if we do this division, we divide F of X by this factor X minus four and we get no remainder. So we just have a polynomial left over. That means that it is a zero. OK. That is indeed a factor of this function. However, in this case, we got a remainder that is nonzero. OK? So our remainder is not zero, which tells us that X equals four is not a zero. All right. So we found that it's not a zero. The second part of the question is asking us if it's not a zero, evaluate F at four. Yeah. Well, you might think we need to do something else. We have to do some more math. But remember the remainder theorem. OK. The remainder theorem tells us that if we take a function F of X and divide it by X minus A, then the remainder is the function app evaluated at A, OK. So we've divided by X minus four. So our a value is four. OK. So the remainder is going to be that function F evaluated at four, right? Exactly what we're looking for. OK. Our remainder is equal to negative 168. OK. So by remainder theorem, and I'm just short forming this, our function F evaluated at four is going to be equal to that remainder. So it's equal to negative 168. All right. So using the synthetic division, we found that X equals four is not a zero of our function F of X. And then F evaluated at four is equal to negative 168. This corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = 2x^3 - 6x^2 -9x + 4; k=1

Key Concepts

Synthetic Division

Polynomial Functions

Evaluating a Function