Question 85751
Start with the given polynomial {{{(2x^4 - 8x^2 + 6)/(2x+2)}}}


First lets find our test zero:


{{{2x+2=0}}} Set the denominator {{{2x+2}}} equal to zero

{{{x=-1}}} Solve for x.


so our test zero is -1



Now set up the synthetic division table by placing the test zero in the upperleft corner and placing the coefficients of the numerator to the right of the test zero.(note: remember if a polynomial goes from {{{2x^4}}} to {{{-8x^2}}} there is a zero coefficient for {{{x^3}}}. This is simply because {{{2x^4 - 8x^2 + 6}}} really looks like {{{2x^4+0x^3+-8x^2+0x^1+6x^0}}}<TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</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 2)

<TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</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>2</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -1 by 2 and place the product (which is -2)  right underneath 0

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</TD></TR><TR><TD></TD><TD></TD><TD></TD><TD>-2</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -2 and 0 to get -2. Place the sum right underneath -2.

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</TD></TR><TR><TD></TD><TD></TD><TD></TD><TD>-2</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-2</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -1 by -2 and place the product (which is 2)  right underneath -8

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</TD></TR><TR><TD></TD><TD></TD><TD></TD><TD>-2</TD><TD>2</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-2</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 2 and -8 to get -6. Place the sum right underneath 2.

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</TD></TR><TR><TD></TD><TD></TD><TD></TD><TD>-2</TD><TD>2</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-2</TD><TD>-6</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -1 by -6 and place the product (which is 6)  right underneath 0

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</TD></TR><TR><TD></TD><TD></TD><TD></TD><TD>-2</TD><TD>2</TD><TD>6</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-2</TD><TD>-6</TD><TD></TD><TD></TD></TR></TABLE>

    Add 6 and 0 to get 6. Place the sum right underneath 6.

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</TD></TR><TR><TD></TD><TD></TD><TD></TD><TD>-2</TD><TD>2</TD><TD>6</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-2</TD><TD>-6</TD><TD>6</TD><TD></TD></TR></TABLE>

    Multiply -1 by 6 and place the product (which is -6)  right underneath 6

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</TD></TR><TR><TD></TD><TD></TD><TD></TD><TD>-2</TD><TD>2</TD><TD>6</TD><TD>-6</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-2</TD><TD>-6</TD><TD>6</TD><TD></TD></TR></TABLE>

    Add -6 and 6 to get 0. Place the sum right underneath -6.

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>2</TD><TD>0</TD><TD>-8</TD><TD>0</TD><TD>6</TD></TR><TR><TD></TD><TD></TD><TD></TD><TD>-2</TD><TD>2</TD><TD>6</TD><TD>-6</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-2</TD><TD>-6</TD><TD>6</TD><TD>0</TD></TR></TABLE>

Since the last column adds to zero, we have a remainder of zero. This means {{{2x+2}}} is a factor of  {{{2x^4 - 8x^2 + 6}}}


Now lets look at the bottom row of coefficients:


The first 4 coefficients (2,-2,-6,6) form the quotient


{{{2x^3 - 2x^2 - 6x + 6}}}

Notice in the denominator {{{2x+2}}}, the x term has a coefficient of 2, so we need to divide the quotient by 2 like this:

{{{(2x^3 - 2x^2 - 6x + 6)/2=x^3 - x^2 - 3x + 3}}}

  

So {{{(2x^4 - 8x^2 + 6)/(2x+2)=x^3 - x^2 - 3x + 3}}}