Question 133344
# 1



Let's simplify this expression using synthetic division



Start with the given expression {{{(8x^4 + 5x^3 + 4x^2 - x + 7)/(x+1)}}}


First lets find our test zero:


{{{x+1=0}}} Set the denominator {{{x+1}}} 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 upper left corner and placing the coefficients of the numerator to the right of the test zero.<TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</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 8)

<TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</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>8</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -1 by 8 and place the product (which is -8)  right underneath the second  coefficient (which is 5)

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-8</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>8</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -8 and 5 to get -3. Place the sum right underneath -8.

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-8</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>8</TD><TD>-3</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -1 by -3 and place the product (which is 3)  right underneath the third  coefficient (which is 4)

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-8</TD><TD>3</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>8</TD><TD>-3</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 3 and 4 to get 7. Place the sum right underneath 3.

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-8</TD><TD>3</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>8</TD><TD>-3</TD><TD>7</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -1 by 7 and place the product (which is -7)  right underneath the fourth  coefficient (which is -1)

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-8</TD><TD>3</TD><TD>-7</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>8</TD><TD>-3</TD><TD>7</TD><TD></TD><TD></TD></TR></TABLE>

    Add -7 and -1 to get -8. Place the sum right underneath -7.

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-8</TD><TD>3</TD><TD>-7</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>8</TD><TD>-3</TD><TD>7</TD><TD>-8</TD><TD></TD></TR></TABLE>

    Multiply -1 by -8 and place the product (which is 8)  right underneath the fifth  coefficient (which is 7)

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-8</TD><TD>3</TD><TD>-7</TD><TD>8</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>8</TD><TD>-3</TD><TD>7</TD><TD>-8</TD><TD></TD></TR></TABLE>

    Add 8 and 7 to get 15. Place the sum right underneath 8.

    <TABLE cellpadding=10><TR><TD>-1</TD><TD>|</TD><TD>8</TD><TD>5</TD><TD>4</TD><TD>-1</TD><TD>7</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-8</TD><TD>3</TD><TD>-7</TD><TD>8</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>8</TD><TD>-3</TD><TD>7</TD><TD>-8</TD><TD>15</TD></TR></TABLE>

Since the last column adds to 15, we have a remainder of 15. This means {{{x+1}}} is <b>not</b> a factor of  {{{8x^4 + 5x^3 + 4x^2 - x + 7}}}

Now lets look at the bottom row of coefficients:


The first 4 coefficients (8,-3,7,-8) form the quotient


{{{8x^3 - 3x^2 + 7x - 8}}}


and the last coefficient 15, is the remainder, which is placed over {{{x+1}}} like this


{{{15/(x+1)}}}




Putting this altogether, we get:


{{{8x^3 - 3x^2 + 7x - 8+15/(x+1)}}}


So {{{(8x^4 + 5x^3 + 4x^2 - x + 7)/(x+1)=8x^3 - 3x^2 + 7x - 8+15/(x+1)}}}


which looks like this in remainder form:

{{{(8x^4 + 5x^3 + 4x^2 - x + 7)/(x+1)=8x^3 - 3x^2 + 7x - 8}}} remainder 15



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






<hr>




# 2




Let's simplify this expression using synthetic division



Start with the given expression {{{(12x^3 + 31x^2 - 17x - 6)/(x+3)}}}


First lets find our test zero:


{{{x+3=0}}} Set the denominator {{{x+3}}} equal to zero


{{{x=-3}}} Solve for x.


so our test zero is -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>12</TD><TD>31</TD><TD>-17</TD><TD>-6</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 12)

<TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>12</TD><TD>31</TD><TD>-17</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -3 by 12 and place the product (which is -36)  right underneath the second  coefficient (which is 31)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>12</TD><TD>31</TD><TD>-17</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-36</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -36 and 31 to get -5. Place the sum right underneath -36.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>12</TD><TD>31</TD><TD>-17</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-36</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-5</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -3 by -5 and place the product (which is 15)  right underneath the third  coefficient (which is -17)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>12</TD><TD>31</TD><TD>-17</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-36</TD><TD>15</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-5</TD><TD></TD><TD></TD></TR></TABLE>

    Add 15 and -17 to get -2. Place the sum right underneath 15.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>12</TD><TD>31</TD><TD>-17</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-36</TD><TD>15</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-5</TD><TD>-2</TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>12</TD><TD>31</TD><TD>-17</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-36</TD><TD>15</TD><TD>6</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-5</TD><TD>-2</TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>12</TD><TD>31</TD><TD>-17</TD><TD>-6</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-36</TD><TD>15</TD><TD>6</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>12</TD><TD>-5</TD><TD>-2</TD><TD>0</TD></TR></TABLE>

Since the last column adds to zero, we have a remainder of zero. This means {{{x+3}}} is a factor of  {{{12x^3 + 31x^2 - 17x - 6}}}


Now lets look at the bottom row of coefficients:


The first 3 coefficients (12,-5,-2) form the quotient


{{{12x^2 - 5x - 2}}}



So {{{(12x^3 + 31x^2 - 17x - 6)/(x+3)=12x^2 - 5x - 2}}}


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