Use synthetic division to perform each division. x^3 - 1 / x-1

Question

Accepted Answer

Hey, everyone here we are asked to perform synthetic division for the expression where we have the quantity of X cubed minus 128 all divided by the quantity of X minus 12. Here we have four answer choice options. Answer choice A X squared plus 2, 12, X minus 144. Answer choice B X squared plus 12, X plus 144. Answer choice C X squared plus 12, X minus 144 plus six divided by the quantity of X minus 12. And finally answer choice D X squared plus 12, X minus 144 minus six divided by the quantity of X minus 12. Now, our first step here is to just rewrite our dividend by showing the coefficients to each of our powers. So we begin with our first term, our highest power, which is X cubed. So we have again X cubed which is missing the coefficient of one. So we have one X cubed, then we want to include the coefficient for the missing terms. So we want to go on to X squared. So we just have plus zero X squared. Since we don't actually have an X squared term, then we also want to include X. So we have plus zero X and finally, we have minus 1728. Now we want to take our divisor and find the zero. So we take the divisor X minus 12 and set it equals to zero to find our zero. So solving for X, we add 12 to both sides. And we see that we have a zero when X is equal to 12. Now with our zero identified and our dividend rewritten, we can begin setting up our synthetic division table. So we place a vertical line and then we want a horizontal line drawn to that crosses through our vital vertical line. And now to the left of the vertical line, we place the value of our zero. So we have the value 12. And now to the right of the vertical line, we place each of the values of our coefficients beginning with the highest X power. So we begin with our first term which is just one. Then we have 04 X squared, then we have 04 X. And lastly, we have have negative 1,728. So now that we have set up our table, we can move on to actually using or applying synthetic division. So the first step is to bring down our first coefficient which is one and we rewrite it underneath the horizontal line. Then we take this value and we multiply it by the zero. So we have one times 12 simplifying, we just get 12. And now we add this product to the coefficient in the second column. So we have zero plus 12, which is just 12. So we write 12 underneath the horizontal line in the second column. Then we repeat this process now taking 12 and multiplying it by 12, this gives us 144. And again, we add this product to the coefficient in the third column. Now, so we have zero plus 144, which is just 144. And we rewrite this underneath the horizontal line. And now again, we take our value 144 multiply it by 12, which gives us 1,728. Now we add this product to our last value here. Our last coefficient. So we have negative 1,728 plus 1,728. These two terms cancel and we just have zero. Now, with the values we obtained using synthetic division, we can write out our quotient. So beginning with our first value, we have one. Well, we want to look to the column to the right of our value of one to find the power of this term. So we have actually one X squared since our second column. Represents our X squared. So we have one X squared. And now moving on to the second value, we have plus 12 and we look to the column to the right. So we have 12 X, then we have plus 144. And lastly the last value we obtained is just zero. Well, this tells us that we just have a remainder of zero. Now, we just want to clean up our solution here, our quotient. So we just have X squared plus 12 X plus 144. And with this solution, we are left with answer choice B 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^3 - 1 / x-1

Key Concepts

Synthetic Division

Polynomials

Remainder Theorem