Divide 6x 4th power-4x 3rd power-27x 2nd power+18x by the binomial x-2.

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Let's simplify this expression using synthetic division
Start with the given expression {{{(6x^4 - 4x^3 - 27x^2 + 18x)/(x-2)}}}
First lets find our test zero:
{{{x-2=0}}} Set the denominator {{{x-2}}} equal to zero
{{{x=2}}} Solve for x.
so our test zero is 2
Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.(note: remember if a polynomial stops at {{{18x^1}}}, there is a zero coefficient for {{{x^0}}}. This is simply because {{{6x^4 - 4x^3 - 27x^2 + 18x}}} really looks like {{{6x^4+-4x^3+-27x^2+18x^1+0x^0}}}<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>
Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 6)
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>
Multiply 2 by 6 and place the product (which is 12) right underneath the second coefficient (which is -4)
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>12</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>
Add 12 and -4 to get 8. Place the sum right underneath 12.
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>12</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD>8</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>
Multiply 2 by 8 and place the product (which is 16) right underneath the third coefficient (which is -27)
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>12</TD><TD>16</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD>8</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>
Add 16 and -27 to get -11. Place the sum right underneath 16.
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>12</TD><TD>16</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD>8</TD><TD>-11</TD><TD></TD><TD></TD></TR></TABLE>
Multiply 2 by -11 and place the product (which is -22) right underneath the fourth coefficient (which is 18)
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>12</TD><TD>16</TD><TD>-22</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD>8</TD><TD>-11</TD><TD></TD><TD></TD></TR></TABLE>
Add -22 and 18 to get -4. Place the sum right underneath -22.
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>12</TD><TD>16</TD><TD>-22</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD>8</TD><TD>-11</TD><TD>-4</TD><TD></TD></TR></TABLE>
Multiply 2 by -4 and place the product (which is -8) right underneath the fifth coefficient (which is 0)
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>12</TD><TD>16</TD><TD>-22</TD><TD>-8</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD>8</TD><TD>-11</TD><TD>-4</TD><TD></TD></TR></TABLE>
Add -8 and 0 to get -8. Place the sum right underneath -8.
<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>6</TD><TD>-4</TD><TD>-27</TD><TD>18</TD><TD>0</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>12</TD><TD>16</TD><TD>-22</TD><TD>-8</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>6</TD><TD>8</TD><TD>-11</TD><TD>-4</TD><TD>-8</TD></TR></TABLE>
Since the last column adds to -8, we have a remainder of -8. This means {{{x-2}}} is <b>not</b> a factor of {{{6x^4 - 4x^3 - 27x^2 + 18x}}}
Now lets look at the bottom row of coefficients:
The first 4 coefficients (6,8,-11,-4) form the quotient
{{{6x^3 + 8x^2 - 11x - 4}}}
and the last coefficient -8, is the remainder, which is placed over {{{x-2}}} like this
{{{-8/(x-2)}}}
Putting this altogether, we get:
{{{6x^3 + 8x^2 - 11x - 4+-8/(x-2)}}}
So {{{(6x^4 - 4x^3 - 27x^2 + 18x)/(x-2)=6x^3 + 8x^2 - 11x - 4+-8/(x-2)}}}
which looks like this in remainder form:
{{{(6x^4 - 4x^3 - 27x^2 + 18x)/(x-2)=6x^3 + 8x^2 - 11x - 4}}} remainder -8
You can use this <a href=http://calc101.com/webMathematica/long-divide.jsp>online polynomial division calculator</a> to check your work

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