Question 109454
Let's simplify this expression using synthetic division



Start with the given expression {{{(x^2 - 5x - 2)/(x+3)}}}


First lets find our test zero:


{{{x+3=0}}} Set the denominator {{{x+3}}} equal to zero


{{{x=-3}}} Solve for x.


so our test zero is -3



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>-3</TD><TD>|</TD><TD>1</TD><TD>-5</TD><TD>-2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><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>-3</TD><TD>|</TD><TD>1</TD><TD>-5</TD><TD>-2</TD></TR><TR><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></TR></TABLE>

    Multiply -3 by 1 and place the product (which is -3)  right underneath the second  coefficient (which is -5)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>1</TD><TD>-5</TD><TD>-2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-3</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD></TD><TD></TD></TR></TABLE>

    Add -3 and -5 to get -8. Place the sum right underneath -3.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>1</TD><TD>-5</TD><TD>-2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-3</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-8</TD><TD></TD></TR></TABLE>

    Multiply -3 by -8 and place the product (which is 24)  right underneath the third  coefficient (which is -2)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>1</TD><TD>-5</TD><TD>-2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-3</TD><TD>24</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-8</TD><TD></TD></TR></TABLE>

    Add 24 and -2 to get 22. Place the sum right underneath 24.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>1</TD><TD>-5</TD><TD>-2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-3</TD><TD>24</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-8</TD><TD>22</TD></TR></TABLE>

Since the last column adds to 22, we have a remainder of 22. This means {{{x+3}}} is <b>not</b> a factor of  {{{x^2 - 5x - 2}}}

Now lets look at the bottom row of coefficients:


The first 2 coefficients (1,-8) form the quotient


{{{x - 8}}}


and the last coefficient 22, is the remainder, which is placed over {{{x+3}}} like this


{{{22/(x+3)}}}




Putting this altogether, we get:


{{{x - 8+22/(x+3)}}}


So {{{(x^2 - 5x - 2)/(x+3)=x - 8+22/(x+3)}}}


which looks like this in remainder form:

{{{(x^2 - 5x - 2)/(x+3)=x - 8}}} remainder 22



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