Use synthetic division to perform each division. x^7+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 to the fourth power minus 1296 all divided by the quantity of X minus six. Here we have four answer choice options. Answer A X cubed plus six X squared plus 36 X minus 216. Answer BX cubed plus six X squared plus 36 X plus 216. Answer CX cubed plus six X squared plus 36 X minus 216 minus two divided by the quantity of X minus six. And answer choice DX cubed plus six X squared plus 36 X plus 216 plus two divided by the quantity of X minus six. So here to begin solving this problem, we first want to identify all of the coefficients of our dividend which again we have X to the fourth power minus 1296. So we want to start with the coefficient of the highest power of X, which here we have as X to the fourth power which has a coefficient of one then moving on to X cubed, we see we do not have an X cued term. So it's coefficient is zero. Then X squared we see is also not present in this expression. So X squared also has a coefficient of zero. And lastly the same for our coefficient of X, we do not see X in our expression. So it's coefficient is also zero. And then lastly our coefficient for X to the zero power or just our constant value here, we have negative 1296. So now we have our coefficients identified and we just want to find our zero from our divisor. So the expression in the denominator, our divisor is X minus six. So we set this expression equal to zero to find our value for X. And so we just add six to both sides and we see X is equal to six. So now with this value for X as well as our coefficients from our expression, we want to set up our format for synthetic division. So we first just draw this vertical line and then a horizontal line which crosses through it. And then to the left of the vertical line, we place the value for X which is six. And then to the right of the vertical line, we place each of our code coefficients of X starting with the highest power of X which again had a coefficient of one. Then we just have 000 and lastly negative 1296. So now we have finished setting up for synthetic division so we can begin performing the process. And the first step is to just bring down the first coefficient which is one and rewrite it underneath the horizontal line. We then take this value of one and multiply it by our value of X which is six. So one time six gives us six, which we add to the coefficient in the second column which is zero. So we have zero plus six giving us six which we rewrite underneath the horizontal line. We then repeat this process taking the value of six and multiplying it by six. This gives us 36 which we add to zero. So we have zero plus 36 in the third column which gives us 36 again and we re rewrite this 36 underneath the horizontal line in its appropriate column. We now take this value repeating the process, we have 36 times six which simplifies to 216. And we add this product to the zero in the fourth column. Zero plus 216 gives us 216. And now repeating this process one more time, we have 216 times six which gives us 1296. Now adding this product to our last columns coming, we have negative 1296 plus 1296 which cancel each other out and give us zero. So now from the values we obtained using synthetic division, we can write out our quotient. So beginning with our first value, we have one. Well, this value is the coefficient of the X to the power to the right column. So this is going to be the coefficient of X cubed. So our first term is one X cube and now looking at our next value we have six and then looking to the right of six, we see it is the column for X squared. So our second term is going to be plus six X squared, then repeating this for the next value of 36 and looking to the column to the right, our third term is going to be plus 36 X. And then from our fourth value, we have a consonant of plus 216. And then our last value here, this zero that we obtained tells us our remainder. So we have our quotient with a remainder of zero. So now we can just rewrite our quotient as X cubed plus six X squared plus 36 X plus 216. And again, we have a remainder of zero which does not need to be included in our final solution. So with this final expression or our final quo, we are left with answer choice B where again, Our quotient is X cubed plus six X squared plus 36 X plus 216. 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^7+1 / x+1

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem