Question 147748

First set up the synthetic division table by placing the test zero (which in this case is -4) in the upper left corner and placing the coefficients of the function to the right of the test zero.<TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>2</TD><TD>-11</TD><TD>-12</TD></TR><TR><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></TR></TABLE>

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)

<TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>2</TD><TD>-11</TD><TD>-12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -4 by 1 and place the product (which is -4)  right underneath the second  coefficient (which is 2)

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>2</TD><TD>-11</TD><TD>-12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -4 and 2 to get -2. Place the sum right underneath -4.

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>2</TD><TD>-11</TD><TD>-12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-2</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -4 by -2 and place the product (which is 8)  right underneath the third  coefficient (which is -11)

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>2</TD><TD>-11</TD><TD>-12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>8</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-2</TD><TD></TD><TD></TD></TR></TABLE>

    Add 8 and -11 to get -3. Place the sum right underneath 8.

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>2</TD><TD>-11</TD><TD>-12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>8</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-2</TD><TD>-3</TD><TD></TD></TR></TABLE>

    Multiply -4 by -3 and place the product (which is 12)  right underneath the fourth  coefficient (which is -12)

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>2</TD><TD>-11</TD><TD>-12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>8</TD><TD>12</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-2</TD><TD>-3</TD><TD></TD></TR></TABLE>

    Add 12 and -12 to get 0. Place the sum right underneath 12.

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>2</TD><TD>-11</TD><TD>-12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>8</TD><TD>12</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-2</TD><TD>-3</TD><TD>0</TD></TR></TABLE>


Since the last column adds to zero, we have a remainder of zero. This means that -4 is a solution of the equation



Now lets look at the bottom row of coefficients:


The first 3 coefficients (1,-2,-3) form the quotient


{{{x^2 - 2x - 3}}}



So {{{(x^3 + 2x^2 - 11x - 12)/(x+4)=x^2 - 2x - 3}}}


Basically  {{{x^3 + 2x^2 - 11x - 12}}} factors to {{{(x+4)(x^2 - 2x - 3)}}}


Now lets solve  {{{x^2 - 2x - 3=0}}}:



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-2) +- sqrt( (-2)^2-4(1)(-3) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-2}}}, and {{{c=-3}}}



{{{x = (2 +- sqrt( (-2)^2-4(1)(-3) ))/(2(1))}}} Negate {{{-2}}} to get {{{2}}}. 



{{{x = (2 +- sqrt( 4-4(1)(-3) ))/(2(1))}}} Square {{{-2}}} to get {{{4}}}. 



{{{x = (2 +- sqrt( 4--12 ))/(2(1))}}} Multiply {{{4(1)(-3)}}} to get {{{-12}}}



{{{x = (2 +- sqrt( 4+12 ))/(2(1))}}} Rewrite {{{sqrt(4--12)}}} as {{{sqrt(4+12)}}}



{{{x = (2 +- sqrt( 16 ))/(2(1))}}} Add {{{4}}} to {{{12}}} to get {{{16}}}



{{{x = (2 +- sqrt( 16 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (2 +- 4)/(2)}}} Take the square root of {{{16}}} to get {{{4}}}. 



{{{x = (2 + 4)/(2)}}} or {{{x = (2 - 4)/(2)}}} Break up the expression. 



{{{x = (6)/(2)}}} or {{{x =  (-2)/(2)}}} Combine like terms. 



{{{x = 3}}} or {{{x = -1}}} Simplify. 



So the answers are:


{{{x=-4}}}, {{{x = 3}}} or {{{x = -1}}}