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) = x^3 - 3x^2 + 4x -4; k=2

Accepted Answer

Hey, everyone in this problem for the given polynomial function, we're asked to perform synthetic division to test if K equals nine is a zero. If it's not to evaluate F F K, the function we're given is F of X equals X cubed minus 18 X squared plus 90 X minus 81. We're given four answer choices. Option A no 171. Option B no, negative 540. Option C no 54 option D E Yes. All right. So we have our function F of X and we wanna know if nine is a zero of this function. Now, in order to do this with synthetic division, what we wanna do is we wanna take X cubed minus 18, X squared plus 90 X minus 81 our function and divide it by X minus nine. OK. So if nine is a zero, that means that X is equal to nine is a zero. OK? And we can rearrange this and we get X minus nine is equal to zero. OK? So we have this factor X minus nine. OK. So we're gonna divide by that factor using synthetic division. Now, if we get a remainder of zero, that means the resulting expression when we do this division is a polynomial. If that's the case, then it's true. This is indeed a zero. OK? Because when we take X minus nine and factor it out, we're left with another polynomial. OK? That is the definition of a zero. All right. So let's do the synthetic division. We're gonna set up our synthetic division table. It's gonna be two rows. We're gonna start with the number on the left hand side outside of our table. And that's just gonna be the number that we're testing as a zero. OK. So we're testing if nine is a zero. And essentially what we're doing is we're taking the zero of the denominator. OK? So we have the denominator is X minus nine, the zero of that is nine. OK? So we've got this number on the outside left of our table. Now we're gonna fill in the first row and the first row is gonna be made up of the coefficients of our function F fa in the numerator starting on the left-hand side. It's gonna be the coefficient corresponding to the highest exponent term. And we're gonna work our way down to the far right where we have the constant term. OK. So our highest exponent term is X cubed and has a coefficient of one. So we get one, we move to our X squared term coefficient negative 18, our X term has a coefficient of 90 and finally our constant term of negative 81. OK. So the first row of our table working from left to right one, negative 18 90 negative 81. Now we're gonna perform the division. We're gonna start on the left hand side and we're gonna bring this one all the way down and underneath our table, when we have a value underneath our table, we multiply it to the value on the left hand side of our table. So we have one multiplied by nine. This gives us nine and we're gonna write it in the next column over in the second row. So now we have a complete column in our second column here. And we're gonna add those values together. We have negative 18 plus nine. This is gonna give us negative nine and we write it down below. Now we have a new value underneath our table negative nine. We're gonna multiply it to this left-hand value of nine, negative nine multiplied by nine gives us negative 81. We write it in the second row of the next column over. Now we have a complete column. So we add that column up 90 plus negative gives us nine. OK. That's a new value underneath. We multiply it to the value on the left nine multiplied by nine gives us 81. Now we have another complete column in this last column and we're gonna add again negative 81 plus 81 gives us zero. And we've done the division. Now we're left with interpreting the results. Now we're gonna draw a dashed line separating the last column on the right hand side from the rest of the table. And this final value underneath our table of zero, this is our remainder. So in this case, our remainder is equal to zero, which tells us that X equals nine is in fact a zero. OK? When we divide our function by the factor X minus nine, we get a remainder of zero. And so that is a zero. Ok. So the correct answer here is option D Yes, it is a zero. 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) = x^3 - 3x^2 + 4x -4; k=2

Key Concepts

Synthetic Division

Polynomial Function

Zero of a Polynomial