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)=5x^3-3x^2+2x-6; k=2

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Hey, everyone in this problem for the following value of K, we're asked to divide F of X by X minus K using synthetic division which will the rate the answer in the form F of X is equal to the factor X minus K multiplied by Q of X plus R. The function we're given is F of X is equal to three X cubed minus six X squared plus five X minus 11. And we're given a K value of three. We have four answer choices. Option A F of X is equal to the factor X minus three multiplied by the function three X squared plus three, X minus 14 plus 31. Option B F of X is equal to the factor X minus three multiplied by the function three X squared minus three X plus 14, All Plus 31. Option CF of X is equal to the factor X - multiplied by the function three X squared plus three X plus 14 minus 31. And option DF of X is equal to the factor X -3 multiplied by the function three X squared plus three X plus 14, all plus 31. So what we're asked to do is we'll take this function three, X Q minus six, X squared Plus five, X -11 and divide it by X minus three. OK. We're asked to divide it by X minus K and we're given a K value of three. So let's set up our synthetic division table on the left hand side, we're gonna start filling out that value. OK? And that value is going to be AK Value three. OK. So this three comes from our K value and it also comes from the root of the denominator. So the denominator in our expression is X -3. If we set this equal to zero, we get that X is equal to three. And so three is the value we use there. I know we wanna fit fill in the top row of our table starting from the left, we're gonna have the coefficient corresponding with the highest exponent term. And we're gonna work our way down all the way on the right with the coefficient of the constant term. So the highest exponent term in our numerator is three and the coefficient corresponding with our X cube term is three. So we have three, then we moved down to our X squared term. The coefficient is negative six To our x term. The coefficient is five. And with our constant terms, the coefficient is negative 11. So that's the top row of our synthetic division table. And now we're gonna start the actual synthetic division on the left-hand side by bringing this three down. And underneath that are taken when we have a value underneath our table, we're gonna multiply it to the value on the side that we started filling at our table with. So we have three multiplied by three, which gives us nine. We're gonna put that in the next column over and just above the bottom of our table. Now we have this complete call up and we're gonna add the values together. So we have negative six plus nine. This gives us three. OK? So under our table, we write that three. And again, we have a value underneath our table, we're gonna multiply that to the value outside of the table on the left. So three multiplied by three again gives us nine. We write it in the next column over and above, we have a complete column. So we add within the column five plus nine gives us 14. That's a new value underneath our table. We multiply that with the number outside of our table on the left 14 multiplied by three, Gives us 42. And again, we have a complete column, we add within the column, We have negative 11 plus 42, which gives us 31. OK. So now we've done our synthetic division and we need to interpret the results. I'm gonna draw a dashed line separating this final column on the right. OK. So this last number on the far right is our remainder. OK. And that's gonna be the R value when we write our function. Yeah. So let's give ourselves some more. So our function F FX Is going to be equal to X -3. OK. That's what we divided by multiplied by the quo of this result. You know how do you write the quota? We're gonna start on the left hand side here. Now we had a function with the highest exponent of three. OK X cubed. That was our highest exponent term. We divided by a function that wasn't linear. OK. We had an X to the exponent one. So we had an exponent three divided by an exponent one. So our new resulting quo is gonna have an exponent of two. Can we reduce that exponent by 1? So starting on the left hand side underneath their table, these are the coefficients that correspond with our quotient. And from the left, it's the highest exponent term. So we know we have the highest exponent of two. So we're gonna have three X squared, then we add, we move along the bottom here. Our next coefficient is three and that's gonna correspond with the next highest exponent term which is going to be X. So we get three X Again, we move along and this last value before our remainder is gonna correspond with our constant term. So we get plus 40. OK. So our function, we can write it as this term X -3 that we divide it by multiplied by the quotient that we get when we divide Plus the remainder of 31. So we found that the correct answer is the factor X minus three multiplied by three X squared plus three, X plus 14. OK. And then we add 31 at the end that corresponds with answer choice. D Thanks everyone for watching. I hope this video helped see you in the next one.

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)=5x^3-3x^2+2x-6; k=2

Verified Solution

Key Concepts

Synthetic Division

Polynomial Remainder Theorem

Quotient and Remainder Form