Use synthetic division to perform each division. (x^4 - 3x^3 - 4x...

Question

Use synthetic division to perform each division. (x^4 - 3x^3 - 4x^2 + 12x) / x-2

Accepted Answer

Hey, everyone here we are asked to perform synthetic division for the problem where we have the quantity of X to the fourth power minus two X cubed minus 17 X squared plus 36 X all divided by the quantity of X minus. Here we have four answer choice options. Answer choice A X cubed plus two, X squared minus nine X. Answer choice B X cubed plus two X squared minus nine X plus two. Answer choice C X cubed plus two X squared minus nine X plus five divided by the quantity of X minus four. And finally answer choice D X cubed plus two X squared minus nine X minus five divided by the quantity of X minus four. So to begin solving this problem, we first want to rewrite our dividend so that we clearly show all of our coefficients. So starting with our first term where we have X to the fourth power. Well, here we are, we are just missing the coefficient, which is just one. So we have one X to the fourth power and now we can just rewrite the next three terms since they all clearly show the coefficient. And they're also descending in order from the highest power term. So writing we have again, one X to the fourth power minus two X cubed minus 17 X squared plus X. And lastly, we want to include plus zero, since we want to include the coefficient for X to the zero power. So now we want to find our zero from the divisor. So we set the divisor X -4 equal to zero. Now solving for X, we just need to add four to both sides. And we see that our zero is when X is equal to four. So now with our zero identified as well as our dividend rewritten clearly, we can begin setting up the table required to perform synthetic division. So for this table, we place a vertical line and then we place a horizontal line that crosses through the vertical line. And on the left side of the vertical line, we place the value of our zero which is four. And then to the right side of the vertical line, we place each of our coefficients for each of our X terms starting with the highest power. So first, our first coefficient is one, then we have negative two, then we have negative 17, then we have 36. And lastly, we just have zero. So now we have properly set up our table, we can go ahead and begin performing synthetic division. So the first step in synthetic division is to just bring down our first coefficient which is one and we rewrite it underneath the horizontal line, then we take this value one and we multiply it by our zero value which is four. So we have one times four which simplifies to just four. And now we want to add the product that we found with the coefficient in this second column. So we have negative two plus four which gives us positive two, which we write in the second column underneath the horizontal line. Then we take this value two and we repeat the process by multiplying two by four. Now simplifying we get eight and now we take the product of eight and add it to the third coefficient. So we have negative 17 plus eight which simplifies to negative nine. Repeating the process. Again, we take negative nine and multiply it by four which gives us negative 36. And then we add this product with the fourth coefficient. So we have 36 plus negative 36 which gives us just zero and repeating one more time. We take zero times four which is just zero, adding zero plus zero, just gives us zero. So our last value is zero and with this last value, this is our remainder. So we have a remainder of zero. So now we have finished performing synthetic division and so we want to write out our quotient using the values that we obtained. So beginning with our first value of one, we want to look at the power from the second column which is X Q. So our first value is going to be one X Q then going to our next term, which is two, we look again at the column to the right of two, which is X squared. So we have plus two X squared, then our next term is negative nine. Again, we look to the column to the right, So we have minus nine X. And lastly, we, with our last term here, we have zero looking to the column to the right, it's X to the zero power. So we just have plus zero. And then again, this last value is our remainder. So it, then we have a remainder of zero. And now we just want to clean up our quotient here. So our final solution becomes X Q plus two X squared minus nine X. And with this solution, we are left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Use synthetic division to perform each division. (x^4 - 3x^3 - 4x^2 + 12x) / x-2

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem