Question 138657

In order to find out if x+2 is a factor of {{{2x^4 + 4x^3 - 3x^2 - 3x - 6}}}, we can use synthetic division



So in this case, the test zero is -2




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>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</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>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</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 -2 by 2 and place the product (which is -4)  right underneath the second  coefficient (which is 4)

    <TABLE cellpadding=10><TR><TD>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</TD><TD>-6</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>2</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</TD><TD>-6</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>2</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -2 by 0 and place the product (which is 0)  right underneath the third  coefficient (which is -3)

    <TABLE cellpadding=10><TR><TD>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 0 and -3 to get -3. Place the sum right underneath 0.

    <TABLE cellpadding=10><TR><TD>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>0</TD><TD>-3</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -2 by -3 and place the product (which is 6)  right underneath the fourth  coefficient (which is -3)

    <TABLE cellpadding=10><TR><TD>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>0</TD><TD>6</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>0</TD><TD>-3</TD><TD></TD><TD></TD></TR></TABLE>

    Add 6 and -3 to get 3. Place the sum right underneath 6.

    <TABLE cellpadding=10><TR><TD>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>0</TD><TD>6</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>0</TD><TD>-3</TD><TD>3</TD><TD></TD></TR></TABLE>

    Multiply -2 by 3 and place the product (which is -6)  right underneath the fifth  coefficient (which is -6)

    <TABLE cellpadding=10><TR><TD>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>0</TD><TD>6</TD><TD>-6</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>0</TD><TD>-3</TD><TD>3</TD><TD></TD></TR></TABLE>

    Add -6 and -6 to get -12. Place the sum right underneath -6.

    <TABLE cellpadding=10><TR><TD>-2</TD><TD>|</TD><TD>2</TD><TD>4</TD><TD>-3</TD><TD>-3</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-4</TD><TD>0</TD><TD>6</TD><TD>-6</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>0</TD><TD>-3</TD><TD>3</TD><TD>-12</TD></TR></TABLE>




Answer:


Since the last column adds to -12, we have a remainder of -12. This means {{{x+2}}} is <b>not</b> a factor of  {{{2x^4 + 4x^3 - 3x^2 - 3x - 6}}}