Use synthetic division to divide ƒ(x) by x-k for the given value ...

Question

Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x) = (x-k) q(x) + r. 
ƒ(x) = 2x^3 + 3x^2 - 16x+10; k = -4

Accepted Answer

Hey, everyone here we are asked to use synthetic division to divide F of X by X minus K. We are then asked to write F of X in the form F of X is equal to the quantity of X minus K times Q of X plus R where Q of X and R are the quotient and the remainder respectively. Here, the given function is F of X is equal to three X cubed plus 19 X squared plus 14, X minus 22. And the given value for K is negative five. Here we have four answer choice options, ABC and D, each of which contain a different function for F of X, but they are each in the requested form. So before we can begin solving, using synthetic division, we first want to identify each of the coefficients given by our given function. So we can start with the highest power of X which is X cubed. So the first coefficient here is going to be three. Then looking at X squared, we have the coefficient of 19 from X, we get the coefficient of 14 and then lastly the coefficient of X to the zeroth power or just our constant value here is negative 22. And then again, our value four k is negative five. So now with our coefficients and our value four k, we can begin setting up for a synthetic division. So we first have to draw a vertical line and then a horizontal line which passes through it into the left of the vertical line. We place the value four K which again is negative five. And then to the right of the vertical line, we place each of our coefficients that we identified starting with the highest power of X. So our first coefficient is going to be three, then we have 19, then 14 and lastly negative 22. So now we have finished setting up for synthetic division. So we can now begin the process. And the first step is to just bring down the first coefficient which is three and rewrite it underneath the horizontal line. We then take this value of three and multiply it by the value of K which is negative five. So three times negative five gives us negative 15. We then take this product and add it to the coefficient in the second column which is 19. So we have 19 plus negative 15 which leaves us with positive four. Now repeating this process, we instead take four and multiply it by negative five, this gives us negative 20 which we add to the coefficient in the third column. So we have 14 plus negative 20 which gives us negative six. Now repeating this process one more time, we have negative six times negative five which gives us positive 30. We then have negative 22 plus 30 which simplifies to positive eight. So now we have finished performing synthetic division. So now we can just write out our quotient and identify our remainder. So first, starting with the remainder, well, the remainder is found from the last value that we obtained using synthetic division. And here the last value is eight. So we have R is equal to eight. And now with the first three values, we can write out our quotient which again is Q of X as stated in our problem statement. So Q of X, the first term is going to have the coefficient of the first value which is three and three is going to be the coefficient of the X term in the column to the right. So the column to the right is X squared. So our first term is going to be three X squared. And now from our next value, which is four, again, we look to the column to the right, which is the column for X. So our second term is going to be plus four X and then from our last value negative six, we just have minus six. So again, our quotient here is three X squared plus four X minus six with a remainder of eight. So now, we want to recall the form that we need to write our function in which is F of X is equal to the quantity of X minus K times Q of X plus R. And so here we already have our Q of X expression and we also have R identified. And so now we just need to recall that our value four K is negative five. And so now we have all the necessary components here. So we can just plug them all into our formula. So we have F of X is equal to the quantity of X minus negative five times Q of X which is the quantity of three X squared plus four X minus six. Then we have plus R so we just have plus eight in our function. Now just simplifying our first set of parentheses, we see our final simplified function is F of X is equal to X plus five or the quantity of X plus five times the quantity of three X squared plus four X minus six plus eight. So now with our final simplified expression for F of X, we are left with answer choice A where again our final function is F of X is equal to the quantity of X plus five times the quantity of three X squared plus four X minus six plus A. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x) = (x-k) q(x) + r. ƒ(x) = 2x^3 + 3x^2 - 16x+10; k = -4