# How do you divide (3x^4-3x^3+x^2 + x – 15) /(x – 3)^2?

Jun 29, 2018

$3 {x}^{2} + 15 x + 64 + \frac{250 x - 591}{x - 3} ^ 2$

#### Explanation:

Given: $\frac{3 {x}^{4} - 3 {x}^{3} + {x}^{2} + x - 15}{x - 3} ^ 2$

Using long division, the denominator becomes: ${x}^{2} - 6 x + 9$

${x}^{2} - 6 x + 9 | \overline{3 {x}^{4} - 3 {x}^{3} + {x}^{2} + x - 15}$

We want to eliminate $3 {x}^{4}$. What times ${x}^{2} = 3 {x}^{4}$? $\text{ } 3 {x}^{2}$

Multiply $3 {x}^{2}$ times each of the terms in the divisor and then subtract:

$\text{ } 3 {x}^{2}$
${x}^{2} - 6 x + 9 | \overline{3 {x}^{4} - 3 {x}^{3} + \text{ } {x}^{2} + x - 15}$
$\text{ } \underline{3 {x}^{4} - 18 {x}^{3} + 27 {x}^{2}}$
$\text{ } 15 {x}^{3} - 26 {x}^{2}$

Bring down the next term from the dividend:

$\text{ } 3 {x}^{2}$
${x}^{2} - 6 x + 9 | \overline{3 {x}^{4} - 3 {x}^{3} + \text{ } {x}^{2} + x - 15}$
$\text{ } \underline{3 {x}^{4} - 18 {x}^{3} + 27 {x}^{2}}$
$\text{ } 15 {x}^{3} - 26 {x}^{2} + x$

We want to eliminate $15 {x}^{3}$. What times ${x}^{2} = 15 {x}^{3}$? $\text{ } 15 x$

Multiply $15 x$ times each of the terms in the divisor and then subtract and then bring down the next dividend term:

$\text{ } 3 {x}^{2} + 15 x$
${x}^{2} - 6 x + 9 | \overline{3 {x}^{4} - 3 {x}^{3} + \text{ "x^2 + " } x - 15}$
$\text{ } \underline{3 {x}^{4} - 18 {x}^{3} + 27 {x}^{2}}$
$\text{ "15x^3 - 26x^2 + " } x$
$\text{ } \underline{15 {x}^{3} - 90 {x}^{2} + 135 x}$
$\text{ } 64 {x}^{2} - 134 x - 15$

We want to eliminate $64 {x}^{2}$. What times ${x}^{2} = 64 {x}^{2}$? $\text{ } 64$

Multiply $64$ times each of the terms in the divisor and then subtract

$\text{ } 3 {x}^{2} + 15 x + 64$
${x}^{2} - 6 x + 9 | \overline{3 {x}^{4} - 3 {x}^{3} + \text{ "x^2 + " } x - 15}$
$\text{ } \underline{3 {x}^{4} - 18 {x}^{3} + 27 {x}^{2}}$
$\text{ "15x^3 - 26x^2 + " } x$
$\text{ } \underline{15 {x}^{3} - 90 {x}^{2} + 135 x}$
$\text{ } 64 {x}^{2} - 134 x - 15$
" "ul(64x^2 - 384x + 576
$\text{ } 250 x - 591$

$\frac{3 {x}^{4} - 3 {x}^{3} + {x}^{2} + x - 15}{x - 3} ^ 2 = 3 {x}^{2} + 15 x + 64 + \frac{250 x - 591}{x - 3} ^ 2$