The remainder theorem indicates that when a polynomial ƒ(x) is di...

Question

The remainder theorem indicates that when a polynomial ƒ(x) is divided by x-k, the remainder is equal to ƒ(k). Consider the polynomial function ƒ(x) = x^3 - 2x^2 - x+2. Use the remainder theorem to find each of the following. Then determine the coor-dinates of the corresponding point on the graph of ƒ(x). ƒ (1)

Accepted Answer

Hey, everyone in this problem for the given polynomial function, we're asked to use the remainder theorem to find F of three. We're told to write the coordinates of the respective point on the graph of the function F of X. The function F FX were given is F FX is equal to X cubed minus five, X squared minus six X plus seven. We're given four answer choices. Option A F at three is negative And the coordinates 3 -29. Option B 29 With the coordinates 3 29, Option C -43 with the coordinates 3 -43 And option D 43 with the coordinates 3 43. All right. So let's recall what remainder theorem tells us. OK. We're told to use the remainder theorem. What does the remainder theorem tell us? Well, remainder theorem tells us that if we divide F of X by X minus A, then the remainder is F evaluated that day. OK. So we want to find F F three F evaluated at three. So in our problem, our a value is going to be equal to three. So we want to take our function F FX, which is X cubed -5, x squared -6 x plus seven And divided by X -A. So divided by X -3. OK. When we do this division, the remainder is going to be the function F evaluated at that value of three, which is what we're looking for. All right. So let's set up our synthetic division table in order to do this division. OK. So we set up our table and we're gonna start with the value on the left side of our table. And that value is going to be our a value. OK? It's the root of the denominator which is the same as our a value. OK? So that's gonna be three and then we're gonna fill out the top row of our table and the top row is gonna be made up of the coefficients of our function F of X in the new murder on the left, we're gonna start with the coefficient corresponding with the highest exponent term. And we're gonna work our way down all the way to the right hand side where we have just the constant term. So the highest exponent we have is three, our X cube term has a coefficient of one. And so the first value in that top row is gonna be one, we move to our X squared term with a coefficient of negative five, then our X term the coefficient of negative six and finally, our constant term of seven. All right. Now we're gonna perform the synthetic division. We're gonna start on the left hand side and we're gonna bring that one down and underneath our table. When we have a value underneath our table, we take it and we multiply it to the value on the left of the table. So we have one multiplied by three that gives us three. We're gonna put it to the right and above in the next column. Now we have a complete column. So we add within the column, we have negative five plus three and this gives us negative too. And we write that down below underneath our table. Now we have negative two, a new value under our table. We're gonna multiply that with the value of three on the side of the table, negative two multiplied by three gives us negative six. We have a complete row. So we're going to add negative six plus negative six Gives Us -12. And we write it down below. We have a new value underneath our table. We multiply it to the value to the left of our table. Negative 12 multiplied by three gives us negative 36. And one last time we have a complete column and we're gonna add, we have seven plus negative 36. This is gonna give us -29. So we've done our synthetic division and now we need to interpret the results if we look at our table I'm gonna add a dash line separating this last column on the right hand side. And the last value on the right hand side underneath our table is going to be a remainder. OK? So in this case, our remainder is equal to tw negative 29. Remember we talked about remainder theorem. When we do this division, the remainder is going to be the function evaluated at our a value of three. So F evaluated at three, it's going to be equal to our remainder which is equal to negative 29. OK. So the function evaluated at three is negative 29. And if we want to write this as a coordinate, OK, we have the X value first which is three. That's the value that we evaluated the function at. OK. Comma the corresponding value of our function F FX which is negative 29. Yeah. So our function evaluated at three is equal to negative 29 the corresponding coordinate is three comma negative 29. This corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Remainder Theorem

Polynomial Functions

Graphing Polynomial Functions