Use synthetic division to perform each division. (5x^4 +5x^3 + 2x^2 - x-3) / x+1

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Accepted Answer

Hey, everyone. Here we are asked to perform synthetic division for the expression four X to the fourth power plus 12 X cubed plus six X squared plus 17 X minus three, all divided by the quantity of X plus three. Here we have four answer choice options. Answer choice, A four X cubed plus six X plus four. Answer choice B four X cubed plus six X minus one answer choice C four, X cubed plus seven X squared plus six X plus four plus two divided by the quantity of X plus three. And finally answer choice D four X cubed plus seven X squared plus six, X minus one minus two divided by the quantity of X plus three. So to begin solving this problem, we first want to take a look at our dividend and we want to write out or identify each of our coefficients beginning with the highest power. Well, here the highest power is four. So beginning with X to the fourth power, we have a coefficient of four. Then moving on to X cubed, we have 12. Then for X squared we have 64 X, we have 17. And for X to the zeroth power or our constant is just negative three. And now we want to find a zero by setting our divisor equal to zero. So our divisor is X plus three and we set it equal to zero. Now, solving for X, we just subtract three from both sides and we get X is equal to negative three. So we have a zero for one, X is equal to negative three. So now we can go ahead and begin setting up our table for synthetic division. So we first write a horizontal or a vertical line and then a horizontal line that crosses through the vertical line. And so to the left of the vertical line, we place the value that we obtained from our zero. So we have negative three on the left side and then to the right side, we place each of our coefficients beginning with the highest power. So our first coefficient on the right side is four, then we have 12 then we have six and then 17 and lastly negative three. So now we have set up our table and we can go ahead and begin performing synthetic division. The first step is to just bring down the first coefficient which is four and rewrite it underneath the horizontal line. Now we take this value four and we multiply it by the value on the left side of the vertical line. So we have four times negative three which gives us negative 12. And now we take this product and add it to the coefficient in the second column. So we have 12 plus three which gives us zero. So we write zero underneath the horizontal line in the second column. And now we take this value of zero and again, we multiply it by negative three. So zero times negative three is just zero. And we take, take this product and add it to the coefficient in the third column. So six plus zero is just six which we write in the third column underneath the horizontal line. Then repeating the process, we take six times negative three which gives us negative 18. We then add this product to the coefficient in the fourth column. So 17 plus negative 18 gives us negative one. And then we repeat this process one more time taking negative one times negative three which gives us positive three. And then we add three to the last coefficient which is negative three. So negative three plus three gives us zero. And now we have finished performing synthetic division. And with these values that we obtained, we can write out our quotient. So beginning with the first value, these values are coefficient. So our first coefficient is four but it is the coefficient of X to some power. And to find this power, we look to the right column of this value which is the second column. And here the second column was X cubed. So we have four X cubed as our first term. And then our next term, we, and again, looking to the column to the right of zero, we have plus zero X squared. Then we have for, for our next term plus zero X and lastly, we have minus one. And then this last term, this last value that we obtained from the synthetic division is our remainder. So here, the value is zero, which tells us that we just have a remainder of zero. So now we just want to clean up our solution. So our final answer is four X cubed plus six X minus one. And this leaves us 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. (5x^4 +5x^3 + 2x^2 - x-3) / x+1

Verified Solution

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem