# How do you evaluate \frac { x ^ { 4} + 4x ^ { 3} - 12x ^ { 2} - 10x + 2} { 2x + 1} ?

May 24, 2018

$.5 {x}^{3} + 1.75 {x}^{2} + 6.875 x - 8.4375 + \frac{10.4375}{2 x + 1}$

#### Explanation:

We need to use long division to find our answer.

Step 1: $2 x$ goes into ${x}^{4}$, $.5 {x}^{3}$ times, so we need to multiply our divisor, $2 x + 1$, by $.5 {x}^{3}$, and subtract that from the dividend, ${x}^{4} + 4 {x}^{3} - 12 {x}^{2} - 10 x + 2$.

$.5 {x}^{3} \left(2 x + 1\right) = {x}^{4} + .5 {x}^{3}$

$\left({x}^{4} + 4 {x}^{3} - 12 {x}^{2} - 10 x + 2\right) - \left({x}^{4} + .5 {x}^{3}\right) = 3.5 {x}^{3} - 12 {x}^{2} - 10 x + 2$

Step 2: $2 x$ goes into $3.5 {x}^{3}$, $1.75 {x}^{2}$ times. Repeat step 1 with these values.

$1.75 {x}^{2} \left(2 x + 1\right) = 3.5 {x}^{3} + 1.75 {x}^{2}$

$\left(3.5 {x}^{3} - 12 {x}^{2} - 10 x + 2\right) - \left(3.5 {x}^{3} + 1.75 {x}^{2}\right) = 13.75 {x}^{2} - 10 x + 2$

Step 3: $2 x$ into $13.75 {x}^{2}$, $6.875 x$ times. Repeat step 1.

$6.875 x \left(2 x + 1\right) = 13.75 {x}^{2} + 6.875 x$

$\left(13.75 {x}^{2} - 10 x + 2\right) - \left(13.75 {x}^{2} + 6.875 x\right) = - 16.875 x + 2$

Step 4: $2 x$ into $- 16.875 x$, $- 8.4375$ times. Repeat step 1.

$- 8.4375 \left(2 x + 1\right) = - 16.875 x - 8.4375$

$\left(- 16.875 x + 2\right) - \left(- 16.875 x - 8.4375\right) = 10.4375$

$2 x$ can't go into 10.4375, so it is a remainder. Taking our divisors and remainder, our answer is:

$.5 {x}^{3} + 1.75 {x}^{2} + 6.875 x - 8.4375 + \frac{10.4375}{2 x + 1}$