Question 1204167
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Answers:
Quotient = <font color=red size=4>6x-5</font>
Remainder = <font color=red size=4>1</font>


Explanation


The other tutors have great approaches.
I'll use synthetic division as another alternative.


The numerator polynomial is 6x^2+31x-29.
The coefficients are placed along the top row of the synthetic division table.


To the left of those coefficients is the test root x = -6, which is derived from solving x+6 = 0.


We have this so far
<table border = "1" cellpadding = "5"><tr><td>-6</td><td>6</td><td>31</td><td>-29</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td></tr></table>
Pull down the leading coefficient 6 to place it in the bottom row.
<table border = "1" cellpadding = "5"><tr><td>-6</td><td>6</td><td>31</td><td>-29</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td>6</td><td></td><td></td></tr></table>
Multiply that value (6) by the test root (-6). 
The result -36 is placed just under the 31.
<table border = "1" cellpadding = "5"><tr><td>-6</td><td>6</td><td>31</td><td>-29</td></tr><tr><td></td><td></td><td>-36</td><td></td></tr><tr><td></td><td>6</td><td></td><td></td></tr></table>
Then we add 31 to -36 to get -5. That result is placed under the -36.
<table border = "1" cellpadding = "5"><tr><td>-6</td><td>6</td><td>31</td><td>-29</td></tr><tr><td></td><td></td><td>-36</td><td></td></tr><tr><td></td><td>6</td><td>-5</td><td></td></tr></table>
The previous two steps are repeated to fill out the last column as shown below
<table border = "1" cellpadding = "5"><tr><td>-6</td><td>6</td><td>31</td><td>-29</td></tr><tr><td></td><td></td><td>-36</td><td>30</td></tr><tr><td></td><td>6</td><td>-5</td><td>1</td></tr></table>
The last value in the bottom row is the remainder.
<font color=red size=4>The remainder is 1</font>


The other values in the bottom row are the coefficients of the quotient.
<font color=red size=4>The quotient is 6x-5</font>


This will mean
(6x^2+31x-29)/(x+6) = 6x-5 remainder 1


We can rewrite that as
{{{(6x^2+31x-29)/(x+6) = 6x-5 + 1/(x+6)}}}


Then we can multiply both sides by the LCD (x+6) to get
{{{6x^2+31x-29 = (x+6)(6x-5) + 1}}}


These claims can be verified using a tool like WolframAlpha or the CAS feature in GeoGebra. 
Many other calculators online offer similar capabilities.


Another way to verify is to expand out the right hand side of the last equation we mentioned.
6x^2+31x-29 = (x+6)(6x-5) + 1
6x^2+31x-29 = x(6x-5)+6(6x-5) + 1
6x^2+31x-29 = (6x^2-5x)+(36x-30) + 1
6x^2+31x-29 = 6x^2-5x+36x-30 + 1
6x^2+31x-29 = 6x^2+31x-29
The answer is confirmed.
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