Question 92329


Start with the given polynomial {{{(x^4 + 10x^3 - 348x - 540)/(x+5)}}}


First lets find our test zero:


{{{x+5=0}}} Set the denominator {{{x+5}}} equal to zero


{{{x=-5}}} Solve for x.


so our test zero is -5



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 {{{10x^3}}} to {{{-348x^1}}} there is a zero coefficient for {{{x^2}}}. This is simply because {{{x^4 + 10x^3 - 348x - 540}}} really looks like {{{1x^4+10x^3+0x^2+-348x^1+-540x^0}}}<TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</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>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</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 -5 by 1 and place the product (which is -5)  right underneath the second  coefficient (which is 10)

    <TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-5</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 -5 and 10 to get 5. Place the sum right underneath -5.

    <TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-5</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>5</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -5 by 5 and place the product (which is -25)  right underneath the third  coefficient (which is 0)

    <TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-5</TD><TD>-25</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>5</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -25 and 0 to get -25. Place the sum right underneath -25.

    <TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-5</TD><TD>-25</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>5</TD><TD>-25</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -5 by -25 and place the product (which is 125)  right underneath the fourth  coefficient (which is -348)

    <TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-5</TD><TD>-25</TD><TD>125</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>5</TD><TD>-25</TD><TD></TD><TD></TD></TR></TABLE>

    Add 125 and -348 to get -223. Place the sum right underneath 125.

    <TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-5</TD><TD>-25</TD><TD>125</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>5</TD><TD>-25</TD><TD>-223</TD><TD></TD></TR></TABLE>

    Multiply -5 by -223 and place the product (which is 1115)  right underneath the fifth  coefficient (which is -540)

    <TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-5</TD><TD>-25</TD><TD>125</TD><TD>1115</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>5</TD><TD>-25</TD><TD>-223</TD><TD></TD></TR></TABLE>

    Add 1115 and -540 to get 575. Place the sum right underneath 1115.

    <TABLE cellpadding=10><TR><TD>-5</TD><TD>|</TD><TD>1</TD><TD>10</TD><TD>0</TD><TD>-348</TD><TD>-540</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-5</TD><TD>-25</TD><TD>125</TD><TD>1115</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>5</TD><TD>-25</TD><TD>-223</TD><TD>575</TD></TR></TABLE>

Since the last column adds to 575, we have a remainder of 575. This means {{{x+5}}} is <b>not</b> a factor of  {{{x^4 + 10x^3 - 348x - 540}}}

Now lets look at the bottom row of coefficients:


The first 4 coefficients (1,5,-25,-223) form the quotient


{{{x^3 + 5x^2 - 25x - 223}}}


and the last coefficient 575, is the remainder, which is placed over {{{x+5}}} like this


{{{575/(x+5)}}}




Putting this altogether, we get:


{{{x^3 + 5x^2 - 25x - 223+575/(x+5)}}}


So {{{(x^4 + 10x^3 - 348x - 540)/(x+5)=x^3 + 5x^2 - 25x - 223+575/(x+5)}}}


which looks like this in remainder form:

{{{(x^4 + 10x^3 - 348x - 540)/(x+5)=x^3 + 5x^2 - 25x - 223}}} remainder 575



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