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)=-3x^3+5x-6; k=-1

Accepted Answer

Hey, everyone. Here we are given for the following value of K divide F of X by X minus K using synthetic division. We are also asked to write the answer in the form F of X is equal to the quantity of X minus K times Q of X plus R. Here we are given the function F of X is equal to negative seven X cubed plus two, X minus eight. And we are given the value for K is negative two. And here we have four answer choice options, ABC and D each which have a different expression for our function F of X. So to begin solving this problem, we first need to set up our table for synthetic division. And so we have this vertical line as well as a horizontal line that crosses through it. And to the left of the vertical line, we need to place our value for K which is negative two. And then to the right of the vertical line, we need to list the coefficients from our function starting with the highest power of X coefficient. So we begin here with the coefficient of X cube since we have X cubed as the highest power of X. So we start with negative seven as the first coefficient, then we move on to X squared. Well, we see we do not have an X squared term. So we just place a zero instead as it's coefficient. And so next we move on to the coefficient of X which is two. And lastly, we have our constant value which is the coefficient of X to the zero power which in this case is negative eight. So now we have set up our table so we can begin applying the method of synthetic division. So the first step is to just bring down the first coefficient and rewrite it underneath the horizontal line. So we bring down negative seven and now we take this value of negative seven and we multiply it by the value of K which is negative two. And now simplifying we get a product of 14. And now we take this product and add it to the coefficient in the second column which is zero. So we have a zero plus 14 which gives us 14 which we rewrite underneath the horizontal line. And then we repeat this process now taking 14 and multiplying it by negative two. Simplifying we get negative 28. And now we add this product to the coefficient in the third column. So we have two plus negative 28 which gives us negative 26. And now we repeat this process one more time, we now have negative 26 times negative two. Simplifying we get positive 52 which we now add to the last coefficient which is negative eight. So we have negative eight plus 52 which simplifies to 44. So now we have finished performing synthetic division. So we now just need to write our quotient which is our Q of X. And so starting with the first value we obtained, this is going to be the coefficient of X squared. So we have negative seven X squared. And now moving next to our next value, we have plus 14 X, Then we have -26. And now our last value here is positive 44. Well, this value is our remainder. So we see that R, our remainder is equal to 44. So now we see we have identified our expression for Q of X as well as our value for R and we are already given the value four K. So we can now write out our function in the form where we have the quantity of X minus K times Q of X plus R. So writing our function in this, this form, we have the quantity of X minus negative two times Q of X which we found as the quantity of negative seven X squared plus 14 X minus 26. And then instead of plus R at the end, we now have plus 44 as 44 was our remainder. So now just quickly simplifying here, we see. Our final function is F of X is equal to the quantity of X plus two times the quantity of negative seven X squared plus 14 X minus plus 44. So with this final function, we see we are left with answer choice. C Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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)=-3x^3+5x-6; k=-1

Key Concepts

Synthetic Division

Polynomial Representation

Remainder Theorem