The polynomial long division page introduced the technique of solving polynomial division sums using long division.
This page will further expand on that introduction, by showing more of the types of polynomial long division with remainder examples than can be encountered.
Polynomial Long Division with Remainder
Examples
(1.1)
( 6x^{\tt{2}} + x \space {\text{--}} \space 4 ) \space \div \space ( 2x + 1 )
Solution
1) [ \frac{6x^{\tiny{2}}}{2x} = 3x ]
\begin{array}{r} 3x \space \space \space \space \space \space \space\space \space\space \space\space\space\space\space\\ 2x + 1 \space|\overline{\space 6x^{\tiny{\tt{2}}} + x \space {\text{--}} \space 4}\\ \end{array}
2) [ 3x \times ( 2x + 1 ) = 6x^2 + 3x ]
\begin{array}{r} 3x \space \space\space\space \space \space \space \space \space\space \space\space \space\space\space\space\space\\ 2x + 1 \space |\overline{\space 6x^{\tiny{\tt{2}}} \space + \space x \space\space {\text{--}} \space 4}\\ {\text{--}} \space \underline{6x^{\tiny{2}} \space + 3x \space\space\space\space\space\space} \\ {\text{-}}2x \space {\text{--}} \space 4\\ \end{array}
3) [ \frac{{\text{-}}2x}{2x} = {\text{-}}1 ]
\begin{array}{r} 3x \space \space \space \space {\text{-}}1 \space \space \space \space \space\space\space\space\space\\ 2x + 1 \space |\overline{\space 6x^{\tiny{\tt{2}}} \space + \space x \space\space {\text{--}} \space 4}\\ {\text{--}} \space \underline{6x^{\tiny{2}} \space + 3x \space\space\space\space\space\space} \\ {\text{-}}2x \space {\text{--}} \space 4\\ \end{array}
4) [ {\text{-}}1 \times ( 2x + 1 ) = {\text{-}}2x \space {\text{--}} \space 1 ]
\begin{array}{r} 3x \space \space \space \space {\text{-}}1 \space \space \space \space \space\space\space\space\space\\ 2x + 1 \space |\overline{\space 6x^{\tiny{\tt{2}}} \space + \space x \space\space {\text{--}} \space 4}\\ {\text{--}} \space \underline{6x^{\tiny{2}} \space + 3x \space\space\space\space\space\space} \\ {\text{-}}2x \space {\text{--}} \space 4\\ {\text{--}} \space\space \underline{{\text{-}}2x \space {\text{--}} \space 1}\\ {\text{-}}3\\ \end{array}
( 6x^{\tt{2}} + x \space {\text{--}} \space 4 ) \space \div \space ( 2x + 1 ) \space = \space 3x \space {\text{--}} \space 1 \space {\text{--}} \space \frac{3}{2x + 1}
(1.2)
( x^{\tt{3}} + 4x^{\tt{2}} \space {\text{--}} \space 5x + 1 ) \space \div \space ( x + 1 )
Solution
1) [ \frac{x^{\tiny{3}}}{x} = x^{\tiny{2}} ]
\begin{array}{r} x^{\tiny{2}} \space \space \space \space \space \space \space\space \space\space \space\space \space\space \space\space \space\space\space\space\space\space\space\space\space\\ x + 1 \space|\overline{\space x^{\tiny{\tt{3}}} + 4x^{\tiny{\tt{2}}} \space {\text{--}} \space 5x + 1}\\ \end{array}
2) [ x^{2} \times ( x + 1 ) = x^3 + x^2 ]
\begin{array}{r} x^{\tiny{2}} \space \space \space \space \space \space \space\space \space\space \space\space \space\space \space\space \space\space\space\space\space\space\space\space\space\\ x + 1 \space|\overline{\space x^{\tiny{\tt{3}}} + 4x^{\tiny{\tt{2}}} \space {\text{--}} \space 5x + 1}\\ {\text{--}} \space \underline{x^{\tiny{3}} + \space x^{\tiny{2}} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space} \\ 3x^{\tiny{2}} \space {\text{--}} \space 5x + 1\\ \end{array}
3/4) [ \frac{3x^{\tiny{2}}}{x} = 3x ] [ 3x \times ( x + 1 ) = 3x^2 + 3x ]
\begin{array}{r} x^{\tiny{2}} \space \space \space 3x \space \space\space \space\space \space\space \space\space \space\space\space\space\space\space\space\space\space\\ x + 1 \space|\overline{\space x^{\tiny{\tt{3}}} + 4x^{\tiny{\tt{2}}} \space {\text{--}} \space 5x + 1}\\ {\text{--}} \space \underline{x^{\tiny{3}} + \space x^{\tiny{2}} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space} \\ 3x^{\tiny{2}} \space\space {\text{--}} \space 5x + 1\\ {\text{--}} \space \underline{3x^{\tiny{2}}+ 3x \space\space\space\space\space\space\space} \\ {\text{-}}8x + 1\\ \end{array}
5/6) [ \frac{{\text{-}}8x}{x} = {\text{-}}8 ] [ {\text{-}}8 \times ( x + 1 ) = {\text{-}}8x \space {\text{--}} \space 8 ]
\begin{array}{r} x^{\tiny{2}} \space \space \space 3x \space \space\space {\text{-}}8 \space\space \space\space\space\space\space\space\space\space\space\\ x + 1 \space|\overline{\space x^{\tiny{\tt{3}}} + 4x^{\tiny{\tt{2}}} \space {\text{--}} \space 5x + 1}\\ {\text{--}} \space \underline{x^{\tiny{3}} + \space x^{\tiny{2}} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space} \\ 3x^{\tiny{2}} \space\space {\text{--}} \space 5x + 1\\ {\text{--}} \space \underline{3x^{\tiny{2}}+ 3x \space\space\space\space\space\space\space} \\ {\text{-}}8x + 1\\ {\text{--}} \space \underline{{{\text{-}}8x \space {\text{--}} \space\space 8} } \\ 9\\ \end{array}
( x^{\tt{3}} + 4x^{\tt{2}} \space {\text{--}} \space 5x + 1 ) \space \div \space ( x + 1 ) \space = \space x^{\tt{2}} + 3x \space {\text{--}} \space 8 + \frac{9}{x + 1}
(1.3)
( 2x^{\tt{3}} + 4x + 5 ) \space \div \space ( x + 2 )
Solution
Here there is no x^{2} term in the dividend, but this isn’t an issue.
It can helpful to write an x^{2} term as 0x^{2}, as this can help to keep the sums structured and tidy.
Though alternatively one can just leave a blank space.
1) [ \frac{2x^{\tiny{3}}}{x} = 2x^{\tiny{2}} ]
\begin{array}{r} 2x^{\tiny{2}} \space \space \space \space \space\space \space\space \space\space \space\space \space\space \space\space\space\space\space\space\space\space\space\\ x + 2 \space|\overline{\space x^{\tiny{\tt{3}}} + 0x^{\tiny{\tt{2}}} + 4x + 5}\\ \end{array}
2) [ 2x^{2} \times ( x + 2 ) = 2x^3 + 4x^2 ]
\begin{array}{r} 2x^{\tiny{2}} \space \space \space \space \space\space \space\space \space\space \space\space \space\space \space\space\space\space\space\space\space\space\space\\ x + 2 \space|\overline{\space x^{\tiny{\tt{3}}} + 0x^{\tiny{\tt{2}}} + 4x + 5}\\ {\text{--}} \space \underline{2x^{\tiny{3}} + 4x^{\tiny{2}} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space} \\ {\text{-}}4x^{\tiny{2}} + 4x + 5\\ \end{array}
3/4) [ \frac{{\text{-}}4x^{\tiny{2}}}{x} = {\text{-}}4x ] [ {\text{-}}4x \times ( x + 2 ) = {\text{-}}4x^2 \space {\text{--}} \space 8x ]
\begin{array}{r} 2x^{\tiny{2}} \space \space \space {\text{-}}4x \space \space \space\space \space\space \space\space\space\space\space\space\space\space\space\\ x + 2 \space|\overline{\space x^{\tiny{\tt{3}}} + 0x^{\tiny{\tt{2}}} + 4x + 5}\\ {\text{--}} \space \underline{2x^{\tiny{3}} + 4x^{\tiny{2}} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space} \\ {\text{-}}4x^{\tiny{2}} + 4x + 5\\ {\text{--}} \space \underline{{\text{-}}4x^{\tiny{2}} \space {\text{--}} \space 8x \space\space\space\space\space\space\space} \\ 12x + 5\\ \end{array}
5/6) [ \frac{12x}{x} = 12 ] [ 12 \times ( x + 2 ) = 12x + 24 ]
\begin{array}{r} 2x^{\tiny{2}} \space \space \space {\text{-}}4x \space \space \space\space 12 \space\space\space\space\space\space\space\space\space\\ x + 2 \space|\overline{\space x^{\tiny{\tt{3}}} + 0x^{\tiny{\tt{2}}} + 4x + 5}\\ {\text{--}} \space \underline{2x^{\tiny{3}} + 4x^{\tiny{2}} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space} \\ {\text{-}}4x^{\tiny{2}} + 4x + 5\\ {\text{--}} \space \underline{{\text{-}}4x^{\tiny{2}} \space {\text{--}} \space 8x \space\space\space\space\space\space\space} \\ 12x + \space\space 5\\ {\text{--}} \space \underline{12x + 24} \\ {\text{-}}19\\ \end{array}
( 2x^{\tt{3}} + 4x + 5 ) \space \div \space ( x + 2 ) \space = \space 2x^{\tt{2}} \space {\text{--}} \space 4x + 12 \space {\text{--}} \space \frac{19}{x + 2}
(1.4)
( 4x^{\tt{3}} + x^{\tt{2}} \space {\text{--}} \space 4x + 5 ) \space \div \space ( x^{\tt{2}} + 3x + 2 )
Solution
Long division can also be used when the divisor polynomial is of a larger degree than 1.
1) [ \frac{4x^{\tiny{3}}}{x^{\tiny{2}}} = 4x ]
\begin{array}{r} 4x \space \space \space \space \space\space \space\space \space\space \space\space \space\space \space\space\space\space\space\space\space\space\space\\ x^{\tiny{2}} + 3x + 2 \space|\overline{\space 4x^{\tiny{3}} + x^{\tiny{2}} \space {\text{--}} \space 4x + 5}\\ \end{array}
2) [ 4x \times ( x^2 + 3x + 2 ) = 4x^3 + 12x^2 + 8x ]
\begin{array}{r} 4x \space \space \space \space \space\space \space\space \space\space \space\space\space\space \space\space\space\space \space\space \space\space\space\space\space\space\space\space\space\\ x^{\tiny{2}} + 3x + 2 \space|\overline{\space 4x^{\tiny{3}} + \space\space\space x^{\tiny{2}} \space\space {\text{--}} \space\space 4x \space + \space 5}\\ {\text{--}} \space \underline{4x^{\tiny{3}} + 12x^{\tiny{2}} + 8x \space\space\space\space\space\space\space\space\space} \\ {\text{-}}11x^{\tiny{2}} \space {\text{--}} \space 12x + \space 5\\ \end{array}
3/4) [ \frac{{\text{-}}11x^{\tiny{2}}}{x^{\tiny{2}}} = {\text{-}}11 ]
[ {\text{-}}11 \times ( x^2 + 3x + 2 ) = {\text{-}}11x^2 \space {\text{--}} \space 33x + 12 ]
\begin{array}{r} 4x \space \space \space {\text{-}}11 \space \space \space\space \space\space\space\space \space\space \space\space\space\space\space\space\space\space\space\space\space\\ x^{\tiny{2}} + 3x + 2 \space|\overline{\space 4x^{\tiny{3}} + \space\space\space x^{\tiny{2}} \space\space {\text{--}} \space\space 4x \space + \space 5}\\ {\text{--}} \space \underline{4x^{\tiny{3}} + 12x^{\tiny{2}} + 8x \space\space\space\space\space\space\space\space\space} \\ {\text{-}}11x^{\tiny{2}} \space {\text{--}} \space 12x + \space 5\\ {\text{--}} \space \underline{{\text{-}}11x^{\tiny{2}} \space {\text{--}} \space 33x \space {\text{--}} \space 12} \\ 21x + 27\\ \end{array}
We can actually stop at this point now with this polynomial long division with remainder example,
as we have a remainder of a lower degree than the original divisor.
( 4x^{\tt{3}} + x^{\tt{2}} \space {\text{--}} \space 4x + 5 ) \space \div \space ( x^{\tt{2}} + 3x + 2 ) \space = \space 4x \space {\text{--}} \space 11 + \large{\frac{21x \space + \space 27}{x^{\tiny{2}} + 3x + 2}}