Question 90353
Let our test zero equal 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.(note: remember if a polynomial goes from {{{1x^4}}} to {{{3x^2}}} there is a zero coefficient for {{{x^3}}}. This is simply because {{{x^4 + 3x^2 + 1}}} really looks like {{{1x^4+0x^3+3x^2+0x^1+1x^0}}}<TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</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>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</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 3 by 1 and place the product (which is 3)  right underneath the second  coefficient (which is 0)

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>3</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 3 and 0 to get 3. Place the sum right underneath 3.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>3</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>3</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 3 by 3 and place the product (which is 9)  right underneath the third  coefficient (which is 3)

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>3</TD><TD>9</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>3</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 9 and 3 to get 12. Place the sum right underneath 9.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>3</TD><TD>9</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>3</TD><TD>12</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 3 by 12 and place the product (which is 36)  right underneath the fourth  coefficient (which is 0)

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>3</TD><TD>9</TD><TD>36</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>3</TD><TD>12</TD><TD></TD><TD></TD></TR></TABLE>

    Add 36 and 0 to get 36. Place the sum right underneath 36.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>3</TD><TD>9</TD><TD>36</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>3</TD><TD>12</TD><TD>36</TD><TD></TD></TR></TABLE>

    Multiply 3 by 36 and place the product (which is 108)  right underneath the fifth  coefficient (which is 1)

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>3</TD><TD>9</TD><TD>36</TD><TD>108</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>3</TD><TD>12</TD><TD>36</TD><TD></TD></TR></TABLE>

    Add 108 and 1 to get 109. Place the sum right underneath 108.

    <TABLE cellpadding=10><TR><TD>3</TD><TD>|</TD><TD>1</TD><TD>0</TD><TD>3</TD><TD>0</TD><TD>1</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>3</TD><TD>9</TD><TD>36</TD><TD>108</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>3</TD><TD>12</TD><TD>36</TD><TD>109</TD></TR></TABLE>

Since the last column adds to 109, we have a remainder of 109. 




So P(3)=109




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