# What does ((x^3+x+3)(x-1))/(x-5)^2 equal?

May 24, 2018

${x}^{2} + 9 x + 66 + \frac{437 x - 165}{{x}^{2} - 10 x + 25}$

#### Explanation:

We'll first want to rewrite the polynomials as single expressions.

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

${x}^{4} - {x}^{3} + {x}^{2} - x + 3 x - 3 = {x}^{4} - {x}^{3} + {x}^{2} + 2 x - 3$

${\left(x - 5\right)}^{2} = \left(x - 5\right) \left(x - 5\right) = {x}^{2} - 10 x + 25$

Now we need to use long division to find our answer.

Step 1: ${x}^{2}$ goes into ${x}^{4}$, ${x}^{2}$ times, so we need to multiply our divisor, ${x}^{2} - 10 x + 25$, by ${x}^{2}$, and subtract that from the dividend, ${x}^{4} - {x}^{3} + {x}^{2} + 2 x - 3$.

${x}^{2} \left({x}^{2} - 10 x + 25\right) = {x}^{4} - 10 {x}^{3} + 25 {x}^{2}$

$\left({x}^{4} - {x}^{3} + {x}^{2} + 2 x - 3\right) - \left({x}^{4} - 10 {x}^{3} + 25 {x}^{2}\right) = 9 {x}^{3} - 24 {x}^{2} + 2 x - 3$

Step 2: ${x}^{2}$ goes into $9 {x}^{3}$, $9 x$ times. Repeat step 1 with these values.

$9 x \left({x}^{2} - 10 x + 25\right) = 9 {x}^{3} - 90 {x}^{2} + 225 x$

$\left(9 {x}^{3} - 24 {x}^{2} + 2 x - 3\right) - \left(9 {x}^{3} - 90 {x}^{2} + 225 x\right) = 66 {x}^{2} - 223 x - 3$

Step 3: ${x}^{2}$ into $66 {x}^{2}$, 66 times. Repeat step 1.

$66 \left({x}^{2} - 10 x + 25\right) = 66 {x}^{2} - 660 x + 1650$

$\left(66 {x}^{2} - 223 x - 3\right) - \left(66 {x}^{2} - 660 x + 1650\right) = 437 x - 1653$

Our three divisors are then added together to find our value, ${x}^{2} + 9 x + 66$. However, we have remainder of $437 x - 162$, so our answer is ${x}^{2} + 9 x + 66 + \frac{437 x - 165}{{x}^{2} - 10 x + 25}$