Remainder theorem:


The remainder when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mrow»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/mrow»«/mfenced»«/mfrac»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»P«/mi»«mfenced»«mi»a«/mi»«/mfenced»«/mrow»«/mstyle»«/math».


Synthetic division:

The result from synthetic division confirms the remainder, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»R«/mi»«/mstyle»«/math», is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»4«/mn»«/mstyle»«/math».



So, this statement can be written as

Using synthetic division, we can determine «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»Q«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»R«/mi»«/mstyle»«/math».


You can verify your division by performing the opposite operation, multiplication.

The division statement «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/mrow»«/mfenced»«/mfrac»«mo»=«/mo»«mi»Q«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mfrac»«mi»R«/mi»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» can be rearranged to make a verification statement. Multiply all terms by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/mrow»«/mfenced»«/mstyle»«/math».


Then, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/mrow»«/mfenced»«/mstyle»«/math» is eliminated on the left side of the equal sign, as well as from the term containing the remainder, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»R«/mi»«/mstyle»«/math», on the right side of the equation.


To verify the result, multiply «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«/mrow»«/mfenced»«/mstyle»«/math» by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»Q«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math» and add «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»R«/mi»«/mstyle»«/math». The result should be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math», the polynomial we started with.