Question 92220


Start with the given polynomial {{{(x^4 - 4x^2 + 7x + 15)/(x+4)}}}


First lets find our test zero:


{{{x+4=0}}} Set the denominator {{{x+4}}} equal to zero


{{{x=-4}}} Solve for x.


so our test zero is -4



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 goes from {{{1x^4}}} to {{{-4x^2}}} there is a zero coefficient for {{{x^3}}}. This is simply because {{{x^4 - 4x^2 + 7x + 15}}} really looks like {{{1x^4+0x^3+-4x^2+7x^1+15x^0}}}<TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</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 1)

<TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</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>1</TD><TD></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 0)

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -4 and 0 to get -4. Place the sum right underneath -4.

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-4</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -4 by -4 and place the product (which is 16)  right underneath the third  coefficient (which is -4)

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>16</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-4</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 16 and -4 to get 12. Place the sum right underneath 16.

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>16</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-4</TD><TD>12</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -4 by 12 and place the product (which is -48)  right underneath the fourth  coefficient (which is 7)

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>16</TD><TD>-48</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-4</TD><TD>12</TD><TD></TD><TD></TD></TR></TABLE>

    Add -48 and 7 to get -41. Place the sum right underneath -48.

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>16</TD><TD>-48</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-4</TD><TD>12</TD><TD>-41</TD><TD></TD></TR></TABLE>

    Multiply -4 by -41 and place the product (which is 164)  right underneath the fifth  coefficient (which is 15)

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>16</TD><TD>-48</TD><TD>164</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-4</TD><TD>12</TD><TD>-41</TD><TD></TD></TR></TABLE>

    Add 164 and 15 to get 179. Place the sum right underneath 164.

    <TABLE cellpadding=10><TR><TD>-4</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>-4</TD><TD>7</TD><TD>15</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>16</TD><TD>-48</TD><TD>164</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>-4</TD><TD>12</TD><TD>-41</TD><TD>179</TD></TR></TABLE>

Since the last column adds to 179, we have a remainder of 179. This means {{{x+4}}} is <b>not</b> a factor of  {{{x^4 - 4x^2 + 7x + 15}}}

Now lets look at the bottom row of coefficients:


The first 4 coefficients (1,-4,12,-41) form the quotient


{{{x^3 - 4x^2 + 12x - 41}}}


and the last coefficient 179, is the remainder, which is placed over {{{x+4}}} like this


{{{179/(x+4)}}}




Putting this altogether, we get:


{{{x^3 - 4x^2 + 12x - 41+179/(x+4)}}}


So {{{(x^4 - 4x^2 + 7x + 15)/(x+4)=x^3 - 4x^2 + 12x - 41+179/(x+4)}}}


which looks like this in remainder form:

{{{(x^4 - 4x^2 + 7x + 15)/(x+4)=x^3 - 4x^2 + 12x - 41}}} remainder 179



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