Question 113803
Let's simplify this expression using synthetic division



Start with the given expression {{{(12x^4 - 67x^3 + 108x^2 - 47x + 6)/(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.<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</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 12)

<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</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>12</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 2 by 12 and place the product (which is 24)  right underneath the second  coefficient (which is -67)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>24</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 24 and -67 to get -43. Place the sum right underneath 24.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>24</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-43</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 2 by -43 and place the product (which is -86)  right underneath the third  coefficient (which is 108)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>24</TD><TD>-86</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-43</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -86 and 108 to get 22. Place the sum right underneath -86.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>24</TD><TD>-86</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-43</TD><TD>22</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 2 by 22 and place the product (which is 44)  right underneath the fourth  coefficient (which is -47)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>24</TD><TD>-86</TD><TD>44</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-43</TD><TD>22</TD><TD></TD><TD></TD></TR></TABLE>

    Add 44 and -47 to get -3. Place the sum right underneath 44.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>24</TD><TD>-86</TD><TD>44</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-43</TD><TD>22</TD><TD>-3</TD><TD></TD></TR></TABLE>

    Multiply 2 by -3 and place the product (which is -6)  right underneath the fifth  coefficient (which is 6)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>24</TD><TD>-86</TD><TD>44</TD><TD>-6</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-43</TD><TD>22</TD><TD>-3</TD><TD></TD></TR></TABLE>

    Add -6 and 6 to get 0. Place the sum right underneath -6.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>12</TD><TD>-67</TD><TD>108</TD><TD>-47</TD><TD>6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>24</TD><TD>-86</TD><TD>44</TD><TD>-6</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-43</TD><TD>22</TD><TD>-3</TD><TD>0</TD></TR></TABLE>

Since the last column adds to zero, we have a remainder of zero. This means {{{x-2}}} is a factor of  {{{12x^4 - 67x^3 + 108x^2 - 47x + 6}}}


Now lets look at the bottom row of coefficients:


The first 4 coefficients (12,-43,22,-3) form the quotient


{{{12x^3 - 43x^2 + 22x - 3}}}



So {{{(12x^4 - 67x^3 + 108x^2 - 47x + 6)/(x-2)=12x^3 - 43x^2 + 22x - 3}}}


You can use this <a href=http://calc101.com/webMathematica/long-divide.jsp>online polynomial division calculator</a> to check your work




Summary:


So our quotient is {{{12x^3 - 43x^2 + 22x - 3}}}



Our remainder is zero



Since our remainder is zero, {{{f(2)=0}}}