Use synthetic division to perform each division. (x^3 + 3x^2 +11x + 9) / x+1

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Hey, everyone here we are asked to perform synthetic division for the expression, the quantity of X cubed plus 15 X squared plus 57 X plus seven all divided by the quantity of X plus seven. Here we have four answer choice options, answer choice A X squared plus A X plus one answer choice B X squared plus A X minus one answer choice C X squared plus eight X plus one plus three divided by the quantity of X plus seven. And finally answer choice D X squared plus A X plus one minus three divided by the quantity of X plus seven. So to begin solving this problem, we first want to identify all of our coefficients from our dividend. So we begin with the highest power of X which we have X cubed. So the coefficient of X cubed is one. Now we can move on to X squared which is 15. Then from X, we have 57 then from X to the zeroth power or the constant value. Here we have positive seven. And now we want to take our divisor and find a zero. So we need to set our divisor X plus seven equal to zero. And now solving for X, we subtract seven from both sides and we find that we have a zero for when X is equal to negative seven. So now we can move on to setting up our table for synthetic division. So first we draw a vertical line and then we have a horizontal line that passes through it. And then to the left of the vertical line, we place the value we obtained from our zero here, which is negative seven. So we have negative seven on the left side and to the right side of the vertical line, we place each of our coefficients beginning with the coefficient of the highest power. So first we have the coefficient of one, then we have 15, then we have 57 lastly, we have seven. So now we can begin performing synthetic division. And so the first step is to just bring down the first coefficient which is one and we rewrite it underneath the horizontal line in the first column. And then moving on to the second column, we want to take our previous value which is one and multiply it by the value on the left side of the vertical line, which is negative seven. So we have one times negative seven which gives us negative seven. And then we want to take this product and add it to the coefficient in the second column. So we have fi negative seven. Which gives us eight which we write underneath the horizontal line in the second column. Now we take this value of eight and again, we multiply it by negative seven. So eight times negative seven gives us negative 56. And we take this product and add it to the coefficient. Now in the third column, so 57 plus negative 50 six is just one which we write underneath the horizontal line in the third column. Now we take one and repeat the process one last time. So we have one times negative seven which gives us negative seven. And we take this product and add it to our last coefficient which is seven, so seven plus negative seven gives us zero. And now we have finished performing synthetic division and from these values that we obtained, we can write our quo. So the first three values that we obtained all are coefficients of sum X. So for the first value we have one well, to find which power of X one belongs to, we look to the column to the right of one. And here we have the column for X squared. So our first term is going to be one X squared. And now moving on to the second value, we have eight and looking to the column to the right of eight, we have X. So we have plus eight X for our second term. And then our next term is just plus one. And then this last value that we obtained from dividing is zero and this value tells us our remainder. So in this case, we have a remainder of zero. So now we just want to rewrite our quotient, cleaning it up a bit. So we have X squared plus eight X plus one as our final answer here, which leaves us 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^3 + 3x^2 +11x + 9) / x+1

Verified Solution

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem