Question 90354
Lets find out if -3 is a zero


We can use synthetic division to find out if -3 is a zero


So 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.<TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><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></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>-3</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><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></TR></TABLE>

    Multiply -3 by 2 and place the product (which is -6)  right underneath the second  coefficient (which is -5)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-6</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -6 and -5 to get -11. Place the sum right underneath -6.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-6</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-11</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -3 by -11 and place the product (which is 33)  right underneath the third  coefficient (which is -4)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-6</TD><TD>33</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-11</TD><TD></TD><TD></TD></TR></TABLE>

    Add 33 and -4 to get 29. Place the sum right underneath 33.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-6</TD><TD>33</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-11</TD><TD>29</TD><TD></TD></TR></TABLE>

    Multiply -3 by 29 and place the product (which is -87)  right underneath the fourth  coefficient (which is 12)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-6</TD><TD>33</TD><TD>-87</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-11</TD><TD>29</TD><TD></TD></TR></TABLE>

    Add -87 and 12 to get -75. Place the sum right underneath -87.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-6</TD><TD>33</TD><TD>-87</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-11</TD><TD>29</TD><TD>-75</TD></TR></TABLE>



Since the last column adds to -75, we have a remainder of -75. This means -3 is <b>not</b> a zero




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Lets find out if 2 is a zero


We can use synthetic division to find out if 2 is a zero


So let our test zero equal 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>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><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></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>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><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></TR></TABLE>

    Multiply 2 by 2 and place the product (which is 4)  right underneath the second  coefficient (which is -5)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>4</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 4 and -5 to get -1. Place the sum right underneath 4.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>4</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-1</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 2 by -1 and place the product (which is -2)  right underneath the third  coefficient (which is -4)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>4</TD><TD>-2</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-1</TD><TD></TD><TD></TD></TR></TABLE>

    Add -2 and -4 to get -6. Place the sum right underneath -2.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>4</TD><TD>-2</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-1</TD><TD>-6</TD><TD></TD></TR></TABLE>

    Multiply 2 by -6 and place the product (which is -12)  right underneath the fourth  coefficient (which is 12)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>4</TD><TD>-2</TD><TD>-12</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-1</TD><TD>-6</TD><TD></TD></TR></TABLE>

    Add -12 and 12 to get 0. Place the sum right underneath -12.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>2</TD><TD>-5</TD><TD>-4</TD><TD>12</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>4</TD><TD>-2</TD><TD>-12</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>2</TD><TD>-1</TD><TD>-6</TD><TD>0</TD></TR></TABLE>



Since the last column adds to zero, we have a remainder of zero. This means 2 is a zero


So the 2 is the only zero