# How do you divide (-18a+3a^3+9+6a^2)div(-3+3a) using synthetic division?

##### 2 Answers
Aug 14, 2018

The remainder is $0$ and the quotient is $= {a}^{2} + 3 a - 3$

#### Explanation:

Divide the dividend and the divisor by $3$

Let's perform the synthetic division

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$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$1$$\textcolor{w h i t e}{a a a a a a a a}$$3$$\textcolor{w h i t e}{a a a}$$- 3$

$\textcolor{w h i t e}{a a a a a a a a a}$_________

$\textcolor{w h i t e}{a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a a}$$- 3$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{0}$

The remainder is $0$ and the quotient is $= {a}^{2} + 3 a - 3$

$\frac{{a}^{3} + 2 {a}^{2} - 6 a + 3}{a - 1} = {a}^{2} + 3 a - 3$

Aug 14, 2018

$\left(3 {a}^{3} + 6 {a}^{2} - 18 a + 9\right) \div \left(3 a - 3\right) = \left({a}^{2} + 3 a - 3\right) + \left(0\right)$

#### Explanation:

Here ,

$\left(- 18 a + 3 {a}^{3} + 9 + 6 {a}^{2}\right) \div \left(- 3 + 3 a\right)$

=>(3a^3+6a^2-18a+9)/(3a-3)=(cancel3(a^3+2a^2-6a+3))/(cancel3(a-1)

$\implies \left({a}^{3} + 2 {a}^{2} - 6 a + 3\right) \div \left(a - 1\right)$

Using synthetic division :

We have , $p \left(a\right) = \left({a}^{3} + 2 {a}^{2} - 6 a + 3\right) \mathmr{and} \text{divisor : } a = 1$

We take , coefficients of $p \left(a\right) \to 1 , 2 , - 6 , 3$

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$\underline{\textcolor{w h i t e}{\ldots}} |$ ul(0color(white)( ........)1color(white)(..........)3color(white)(.....)-3
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We can see that , quotient polynomial :

$q \left(a\right) = {a}^{2} + 3 a - 3 \mathmr{and} \text{the Remainder} = 0$

Hence ,

$\left({a}^{3} + 2 {a}^{2} - 6 a + 3\right) \div \left(a - 1\right) = \left({a}^{2} + 3 a - 3\right) + \left(0\right)$

Multiplying numerator and denominator of LHS by $3$

$\frac{3 \left({a}^{3} + 2 {a}^{2} - 6 a + 3\right)}{3 \left(a - 1\right)} = \left({a}^{2} + 3 a - 3\right) + \left(0\right)$

$\frac{3 {a}^{3} + 6 {a}^{2} - 18 a + 9}{3 a - 3} = \left({a}^{2} + 3 a - 3\right) + \left(0\right)$