Use synthetic division to perform each division.

Question

\frac{\frac{1}{3} x^{3} - \frac{2}{9} x^{2} + \frac{2}{27} x - \frac{1}{81}}{x - \frac{1}{3}}

Accepted Answer

Hey, everyone. Here we are asked to perform synthetic division for the expression, the quantity of negative seven X cubed plus 37 X squared plus 17 X plus 54 all divided by the quantity of X minus six. Here we have have four answer choice options. Answer choice, A negative seven X squared minus five, X minus 13 minus 24 divided by the quantity of X minus six. Answer choice B negative seven X squared minus five, X minus 13 plus 24 divided by the quantity of X minus six. Answer choice C negative seven X squared minus five X minus 13. And finally answer choice D negative seven X squared plus five X minus 13. To begin this problem, we first want to find our zero using our divisor. So we just set our divisor X minus six equal to zero and solve for X. So solving for X, we just add six to both sides of the equation and we get a zero for when X is equal to six. And now we just want to identify all of the values of our coefficients starting with the highest power of X. So our highest power is three. So the coefficient of X cubed is negative seven. Then for X squared, we have 37 then for X, we have 17. And lastly, we have 54 4 X to the zeroth power. So with all of our coefficients identified and our zero value identified as well, we can go ahead and start setting up our table for a synthetic division. So we begin by drawing a vertical line and then a horizontal line that crosses through it and to a, to the left of the vertical line, we place the value of our zero, the value of X here which is six. And then to the right of the vertical line, we place each of our coefficients in order of the highest power. So our first coefficient is negative seven, then we have 37 then we have and lastly, we have 54. Now we have our table set up so we can start performing synthetic division. And we first need to bring down our first coefficient with which is negative seven and we rewrite negative seven below the horizontal line in our first column. Now we take this value of negative seven and we multiply it by our zero value which is six. So we have negative seven times six which is equal to negative 42. Then we product and add it to the coefficient in the second column. So we have 37 plus negative simplifying we get negative five. Then again, we take this value negative five and multiply it by six, negative five times six gives us negative 30. And we add this product to the coefficient in the third column. Now, so we have 17 plus negative 30 which gives us negative 13. Then we repeat this process one more time taking negative 13 times six which gets us negative 78. Now we add this product with the last coefficient. So we have 54 plus negative 78 which simplifies to negative 24. Now that we have performed synthetic division, we can take each of these values and write out our quotient. So starting with our first value of negative seven, we have negative seven as the coefficient of X to some power. And to find out what power X is raised to, in this case, we look at the column to the right of the value and the column to the right we have as X squared. So our first value negative seven, we have negative seven X squared. Then for the next value again, we look to the column to the right which is X. So we have minus five X then for the next term we have negative 13 and this just gives us minus 13 X to the zeroth power which is just one. So we just have minus 13. And then this last value we have negative 24 is the remainder. So we have a remainder of negative 24. So now we just want to rewrite our remainder value here by taking the value of negative 24 dividing it by the original divisor. So now just rewriting our quotient, we have negative seven X squared minus five X minus 13. And now adding our remainder, we have plus negative 24. So we just have minus 24 divided by the quantity of X minus six. And now with this solution, we are left here with answer choice B where again we have negative seven X squared minus five, X minus 13 plus 24 divided by the quantity of X minus six. 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. $\frac{\frac{1}{3} x^{3} - \frac{2}{9} x^{2} + \frac{2}{27} x - \frac{1}{81}}{x - \frac{1}{3}}$

Key Concepts

Synthetic Division

Polynomial Coefficients and Terms

Division by a Linear Binomial of the Form x - c

Use synthetic division to perform each division. 13x3−29x2+227x−181x−13\frac{\frac{1}{3}x^3 - \frac{2}{9}x^2 + \frac{2}{27}x - \frac{1}{81}}{x - \frac{1}{3}}

Key Concepts

Synthetic Division

Polynomial Coefficients and Terms

Division by a Linear Binomial of the Form x - c

Use synthetic division to perform each division. $\frac{\frac{1}{3} x^{3} - \frac{2}{9} x^{2} + \frac{2}{27} x - \frac{1}{81}}{x - \frac{1}{3}}$