Question #c7b2b

Mar 12, 2017

The remainder is $= - 3$ and the quotient is $= 3 {y}^{2} - 7$

Explanation:

You can perform a long division

$\textcolor{w h i t e}{a a a a}$$15 {y}^{3} - 27 {y}^{2} - 35 y + 60$$\textcolor{w h i t e}{a a a a}$$|$$5 y - 9$

$\textcolor{w h i t e}{a a a a}$$15 {y}^{3} - 27 {y}^{2}$$\textcolor{w h i t e}{a a a a a a a a a a a a a a}$$|$$3 {y}^{2} - 7$

$\textcolor{w h i t e}{a a a a a a a a}$$0 - 0 - 35 y + 60$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$- 35 y + 63$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a}$$0 - 3$

The remainder is $= - 3$ and the quotient is $= 3 {y}^{2} - 7$

You can check by doing a multiplication

$15 {y}^{3} - 27 {y}^{2} - 35 y + 60 = \left(5 y - 9\right) \left(3 {y}^{2} - 7\right) - 3$

Mar 12, 2017

${y}^{2} + \frac{18}{5} y - \frac{13}{25} + \frac{1383}{125 y - 225}$

Same as:

${y}^{2} + \frac{18}{5} y - \frac{13}{25} + \frac{1383}{25 \left(5 y - 9\right)}$

Explanation:

Given$\text{ } \left(5 {y}^{3} - 27 {y}^{2} - 35 y + 60\right) \div \left(5 y - 9\right)$

$\text{ } 5 {y}^{3} - 27 {y}^{2} - 35 y + 60$
$\textcolor{m a \ge n t a}{{y}^{2}} \left(5 y - 9\right) \to \text{ "ul(5y^3-9y^2) larr" Subtract}$
$\text{ } 0 - 18 {y}^{2} - 35 y + 60$
$\textcolor{m a \ge n t a}{\frac{18}{5} y} \left(5 y - 9\right) \to \text{ "ul(color(white)(.)18y^2-162/5y) larr" Subtract}$
$\text{ } 0 - \textcolor{w h i t e}{.} \frac{13}{5} y + 60$
$\textcolor{m a \ge n t a}{- \frac{13}{25}} \left(5 y - 9\right) \to \text{ "ul(-13/5y+117/25) larr" Subtract}$
$\textcolor{m a \ge n t a}{\text{ Remainder } \to + \frac{1383}{25}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{m a \ge n t a}{{y}^{2} + \frac{18}{5} y - \frac{13}{25} + \left(\frac{1383}{25} \div \left(5 y - 9\right)\right)}$

${y}^{2} + \frac{18}{5} y - \frac{13}{25} + \frac{1383}{125 y - 225}$