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

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Hey, everyone. Here we are asked to perform synthetic division for the expression, the quantity of X to the fifth power plus six X to the fourth power plus eight X cubed plus five X squared plus 17 X minus 10, all divided by the quantity of X plus four. Here we have four answer choice options. Answer choice A X to the fourth power plus two X cubed minus eight X squared plus five, X minus three. Answer choice BX to the fourth power plus two X cubed plus eight X squared plus five X minus three. Answer choice CX to the fourth power plus two X cubed plus five, X minus three minus two divided by the quantity of X plus four. Answer choice DX to the fourth power plus two X cubed plus five X minus three plus two by the quantity of X plus four. So to begin solving this problem, we first want to list out all of our coefficients starting with the highest power of X. So our highest power here is five. So we can begin with the coefficient of X to the fifth power, which in our case is just one and then we can move on to X to the fourth power which is six. Then for X cubed, we have a coefficient of eight. Then for X squared, we have 54, X, we have 17 and four X to the zero power or in other words, our constant here is just negative 10. So that's our last coefficient. And now our next step is to just take our divisor and set it equal to zero to find a zero. So we have X plus four, our divisor equal to zero. Now solving for X, we subtract four from both sides and we get a zero for when X is equal to negative four. Now with our zero identified as well as our coefficients from the dividend, we can go ahead and set up our table so we can then perform synthetic division. So first we place a vertical line and then a horizontal line that crosses through it into the left of the vertical line. We place the value of the zero that we found, which is negative four. And then to the right side of the vertical line, we place each of our co coefficients beginning with the coefficient of the highest power. So the first coefficient we put down is one, then we have 685, 17 and negative 10. So now we have set up our table and we can go ahead and begin performing synthetic division. So the first step is to just bring down our first coefficient which is one and rewrite it underneath the horizontal line in the first column. And then we take this value of one and we multiply it by our value on the left side of the vertical line, which is negative four. So we have one times negative four which gives us negative four. Now we take this value and add it to the coefficient in the second column. So we have six plus negative four which gives us positive two, which we write in the below the horizontal line in the second column. Then we take this value of two and we again multiply it by negative four, this gives us negative eight and we take this product and add it to the coefficient in the third column. So we have eight plus negative eight which is just zero and we write zero underneath the horizontal line in the third column. Now just repeating the process, we now take zero times negative four which just gives us zero. Then we have five plus zero which is just five. Then we take the value of five and we multiply it again by negative four which gives us negative 20. Then we have 17 plus negative 20 which is negative three. And lastly, we repeat this step one more time. So we have negative three times negative four which is positive 12. Now we have negative 10 plus 12 which gives us two. So now we have finished performing synthetic division. And with each of these values that we obtained, we can go ahead and write our quotient. So to find our quotient, we can begin with our first value here, which is one and these values are the coefficients of X. So the first value one is going to be the coefficient of the X to the power of the, the next column, the column to the right. So we have one X squared or sorry X to the fourth power. Then moving on to our next value of two. Again, we look to the column to the right. So we have plus two X cubed. Then for our next value of zero, we have plus zero X squared. Next we have plus five X and then for our last value, we have minus three. And then for the last value that we obtained here from our table is just positive two. Well, this positive two just tells us that we have a remainder of two. So now we just want to rewrite our quotient just cleaning it up a bit. And then we want to rewrite our remainder by taking the value of two and dividing it by the original divisor. So now rewriting, we have X to the fourth power plus two, X cubed plus five, X minus three plus our remainder here, which is two divided by the quantity of X plus four. And now with our solution, we are left with answer choice D 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^5 + 3x^4 + 2x^3 + 2x^2 + 3x+1) / x+2

Verified Solution

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem