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

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) = x³ - 2x² - x+2. Use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of ƒ(x). ƒ (-2)

Accepted Answer

Hey, everyone in this problem for the given polynomial function, we're asked to use the remainder theorem to find F at negative four. We're told to write the coordinates of the respective point on the graph of the function F FX. The function we're given is F of X is equal to X cubed minus four, X squared minus three, X plus five. We're given four answer choices. Option A 111 with the coordinate negative four comma 111. Option B -111 with the Coordinate -4, -111. Option C negative 121 the coordinate negative four comma negative 121. And option D 121 with the coordinate negative four comma 121. Now let's recall what remainder theorem tells us. OK. We're asked to use remainder theorem. Now, remainder theorem tells us that if we divide R function F of X by a factor X minus A, then the remainder is equal to F evaluated at A. So in our case, we wanna find f at negative four Which tells us that our a value is going to be equal to negative four. So we want to take our function F of X which is X cubed minus four, X squared minus three, X plus five and divide it by X minus negative four. Now, when we perform this division, the remainder is going to be that value F at negative four that we're looking for. Now we're dividing polynomials. OK. Whichever way you'd like to do this, you can um synthetic division is gonna be the most efficient way. So we're gonna work with that. OK. We're gonna start by just writing out our synthetic division table. Now, the value on the left-hand side of our table is given by the root of the denominator, which in this case is gonna be equal to our A value. OK? So our, a value is negative four. So that's the value on the left hand side of our table. Then we're gonna fill in the top row of our table with the coefficients corresponding to our function F of X. In the numerator, we're gonna start on the left hand side with the coefficient corresponding to the highest exponent term. Now the highest exponent term in our function F of X is X cubed and it has a coefficient of one. So that's the first number in the next highest exponent. OK? We had cubed. So now we have X squared With a coefficient of -4 and we're just working our way to the right in the top row of our table. Next, we have an X term which with the coefficient of negative three. So we write negative three and finally, our constant term of five. OK. So we have our top row, one, negative four, negative 35. In order to perform synthetic division, we're gonna start on the left. We're gonna bring this one all the way down and under our table. And when we have a value underneath our table, we're gonna multiply it to the value on the left of our table. So we have one multiplied by negative four. This gives us -4 And we write it in the 2nd row of the next column over. And now we have a complete column. We have negative four and negative four and we're gonna add within the column. So negative four plus negative four gives us negative eight. Now we write that down below and we have a new value underneath our table value underneath our table. We've multiplied to the value on the left of our table - multiplied by -4. And that gives us 32. We write it In the next column over in that 2nd row. And again, we have a complete column. So we add within our column, negative three plus 32 gives us 29. That's a new value underneath our table. We multiply it to the left of our table. 29 multiplied by -4 gives us -116. And once again, we have a complete column, we add five plus negative gives us negative 111. So now we've finished our synthetic division and we're left with simplifying the result. OK? Or just interpreting the result, we don't even need to simplify it, we need to interpret. OK. So let's draw a dashed line separating this last column in our synthetic division. And this last value on the right hand side underneath their table is a remainder. So in this case, the remainder is negative may now recall what we mentioned about the remainder theorem. The remainder is going to be equal to that function F evaluated at our A value. OK? So F at negative four in this case, because our, a value is negative four is equal to the remainder Which we found is equal to -111. OK. So our remainder is negative 111. The function evaluated at negative four is negative 111. And we're left with just writing the coordinate at this point. OK. Now the coordinate of the point we're gonna have the X value first and then the Y value. Now the X value is negative four. That's the X value we wanted to evaluate the function at. So our point or coordinate is gonna be negative four comma and then the corresponding F of X value OK, we found that F at negative four is equal to negative 111. So our point is negative four comma negative 111. If we compare this to our answer choices, we see that this matches with answer choice B so the correct answer was the function evaluated at negative four is negative 111. And we have the corresponding coordinate negative four comma negative 111. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Remainder Theorem

Polynomial Evaluation

Coordinates on the Graph of a Function