# How do you divide \frac { 3x ^ { 4} - 16x ^ { 3} - 4x ^ { 2} + 46x + 7} { x - 5}?

Nov 3, 2017

$\frac{3 {x}^{4} - 16 {x}^{3} - 4 {x}^{2} + 46 x + 7}{x - 5} = 3 {x}^{3} - {x}^{2} - 9 x + 1 + \frac{12}{x - 5}$

#### Explanation:

The fastest way to do this is with "synthetic division". You will also need to know how to do "polynomial (aka algebraic) long division", and it is much more useful to know since it works in every situation. Synthetic division only works if you have a denominator of the form $\left(x \pm n\right)$ . It also seems a little magical, and long division doesn't.

Step 1:
That being said, for synthetic division, you draw a box, like in the picture below, and write the coefficients of your terms across the top. MAKE SURE YOUR POLYNOMIAL IS IN STANDARD FORM FIRST!! (Where the biggest variable is first, then the next biggest...then finish with the constant term with no variable) You may want to put another little boxed-off part at the end of the bottom; this is where the remainder will go.

Outside the box, write the negative of the numeric term in the
denominator. Here, since we have $\left(x - 5\right)$, the negative of $- 5$ is just $5$, so we use that. Then bring down the first coefficient and write it under the box.

Step 2:
Multiply that number you just brought down by the number to the left of the box. Write it in the next column and add. $5 \cdot 3 = 15$, so write $15$ under $- 16$ in the second column.

Step 3:
Repeat Step 2 until you've finished multiplying and adding everything. $5 \cdot - 1 = - 5$, and $- 4 \pm 5 = - 9$...

Step 4:
Dividing the first term of your numerator by the first term of your denominator tells you that your answer will begin with $\frac{3 {x}^{4}}{x} = 3 {x}^{3}$.
Now look at the numbers along the bottom of the box you just finished working with, and use that first number as the first coefficient in your answer. The second number is the second coefficient, and continue in descending order.

That last number in the little boxed-off part is the remainder. The remainder is always written above the original denominator.
$\frac{3 {x}^{4} - 16 {x}^{3} - 4 {x}^{2} + 46 x + 7}{x - 5} = 3 {x}^{3} - {x}^{2} - 9 x + 1 + \frac{12}{x - 5}$