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^4 + 4x^3 - 10x^2 + 15; k = -1

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 equation and the remainder respectively. Here, our given function is F of X is equal equal to seven X to the fourth power plus 23 X cubed plus X squared minus 12 X plus 10. And our giving value for K is negative three. Here we have four answer choice options, ABC and D, each of which contain a different function for F of X that is rewritten in the required form. So here, before we begin performing synthetic division, we first want to identify each of the coefficients from our given function starting with the coefficient of the highest power of X. So here we see that the highest power is four. So we can begin with the coefficient of X to the fourth power which is seven. Now moving on to X Q, we have the coefficient of 23. From X squared, we have one from X, we have negative 12. And from our coefficient of X to the zeroth power or just our constant value. Here, we have 10. And then again, remembering that our value four K is negative three. So now with our value four K, as well as all of our coefficients clearly identified, we can begin setting up to perform synthetic division. So we need a vertical line and then a horizontal line that passes through it. And to the left of the vertical line, we need to place the value for K which is negative three. And then to the right of the vertical line, we place each of the coefficients of X starting with the highest power of X. So we begin with our value of seven, then we choose 23 then the next coefficient is one, we then have negative 12 and lastly we have 10. And so now we have finished setting up. So we can begin performing synthetic division. And the first step is to just bring down the first coefficient which is seven, we then take this value of seven and multiply it by the value of K which is negative three. And now taking this product, we get negative 21 then we add this product to the coefficient in the second column. So we have 23 plus negative 21 which gives us value of positive two. We then take this value of two and repeat the process by multiplying it by negative three. Simplifying we get negative six which we add to the coefficient in the third column. So we now have one plus negative six which gives us negative five. Repeating this process. Again, we now take negative five times negative three which gives us positive 15. We then have negative 12 plus 15 which simplifies to positive three, repeating this process one last time we take three and multiply it by negative three, which gives us negative nine, we then add 10 plus negative nine, which gives us positive one. And so now we have finished performing synthetic division. So we just need to now write our quotient and our remainder. Well, from our problem statement, we know our quotient is identified as Q of X. So now writing out our quotient, we begin with the first term that we obtained from synthetic division which is seven, well, seven is going to be the coefficient of the column to the right, which is X cubed. So our first term is going to be seven X cubed. Now looking at our next value that we obtained, which is two, we again look to the column to the right of this value, which is the column for X squared. So our next term is going to be plus two X squared, then we will have minus five X and lastly plus three from our second to last value. And our very last value is one and this value is going to be our remainder value. So we have our quotient with a remainder of one. And so now we have identified Q of X and our remainder. So we now need to recall the form which we need to write F of X. In, in this form, we had F of X is equal to the quantity of X minus K times Q of X plus R. So again, we already have Q two of X and we have R, we just need to recall that K is negative three. So now we have each of the required components to rewrite our function in this form. So we see we have F of X is equal to the quantity of X minus negative three time times Q of X which is the quantity of seven X cubed plus two, X squared minus five X plus three. And then we have plus R from the formula. So we just have plus one in our function. Now just simplifying the first set of parentheses. Here we see our final simplified function is F of X is equal to the quantity of X plus three times the quantity of seven X cubed plus two, X squared minus five X plus three and then plus one. So with this final function F of X, we are left with answer choice C where again, we have F of X is equal to the quantity of X plus three times the quantity of seven X cubed plus two X squared minus five X plus three plus one. 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^4 + 4x^3 - 10x^2 + 15; k = -1

Key Concepts

Synthetic Division

Polynomial Remainder Theorem

Quotient and Remainder Form