Pre-Calculus: Polynomial Long Division

Example

In this example we are given the following info rational expression In other words, a rational expression consists of a fraction in which both the numerator (top) and denominator (bottom) are polynomials. \( \frac{2x^4-21x^2+9x+5}{x+3} \) to solve. Here is a guided example showing the steps for polynomial long divison.

• Setup the equation as follows:

\begin{array}{rll} x+3 \enclose{longdiv}{2x^4+0x^3-21x^2+9x+5}\kern-.2ex \\[-3pt] \end{array}

Notes:

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• The info dividend The dividend is the expression which is to be divided. This is the top part of a fraction. \begin{array}{rl} \frac{\color{DodgerBlue}{3x^3}}{x^2}=3x \end{array} goes under the divison bracket. Ensure the the polynomial is in info descending order Descending order means that the highest degree terms come first. \begin{array}{rl} x^2+3x-4 \end{array} and add placeholders for any missing info degree The degree of a term is the power to which the variable/number is raised. \begin{array}{rl} 5x^{\color{DodgerBlue}{2}} \end{array} terms.
• The info divisor The divisor is the expression which divides the dividend. This is the bottom part of a fraction. \begin{array}{rl} \frac{3x^3}{\color{DodgerBlue}{x^2}}=3x \end{array} goes to the left of the divison bracket.

Now that we have the equation setup we can perform the following 4 steps repeatedly until we are left with zero or a remainder.

1. Divide the current info leading term The leading term is the first term when the polynomial is written in decending order or standard form. \begin{array}{rl} {\color{DodgerBlue}{x^2}}+3x-4 \end{array} in the dividend by the leading term of the divisor.

\begin{array}{rl} \frac{2x^4}{x+3}=2x^3 \end{array}

Notes:

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• We can add the quotient ontop of the term with the same degree within the division bracket.

\begin{array}{rll} 2x^3 \phantom{000000000000000} && \\[-3pt] x+3 \enclose{longdiv}{2x^4+0x^3-21x^2+9x+5}\kern-.2ex \\[-3pt] \end{array}

2. Distribute the quotient from the operation for all terms in the divisor.

\begin{array}{rl} 2x^3(x+3)=2x^4+6x^3 \end{array}

3. Subtract the product polynomial from the initial dividend:

\begin{array}{rll} 2x^3 \phantom{000000000000000} && \\[-3pt] x+3 \enclose{longdiv}{\cancel{2x^4}+0x^3-21x^2+9x+5}\kern-.2ex \\[-3pt] \underline{-(2x^4+6x^3)} \phantom{0000000000000} \\[-3pt] 0x^4-6x^3 \phantom{00000000000000} \\[-3pt] \end{array}

4. Bring down the next term of the dividend polynomial.

\begin{array}{rll} 2x^3 \phantom{00000000000000} && \hbox{(Calculations - Steps 2 & 3)} \\[-3pt] x+3 \enclose{longdiv}{\cancel{2x^4}+0x^3-21x^2+9x+5}\kern-.2ex \\[-3pt] \underline{-(2x^4+6x^3)} \phantom{000} \nabla \phantom{000000000} && 2x^3(x+3)=2x^4+6x^3 \\[-3pt] -6x^3-21x^2 \phantom{00000000} \\[-3pt] \end{array}

• We can now repeat steps 1-4 again: (This time our leading coefficient of our dividend polynomial is \(-6x^3\))

\begin{array}{rll} 2x^3-6x^2 \phantom{0000000000} && \hbox{(Calculations - Steps 2 & 3)} \\[-3pt] x+3 \enclose{longdiv}{\cancel{2x^4}+\cancel{0x^3}-21x^2+9x+5}\kern-.2ex \\[-3pt] \underline{-(2x^4+6x^3)} \phantom{00000} \nabla \phantom{000} \nabla \phantom{0000} && 2x^3(x+3)=2x^4+6x^3 \\[-3pt] -6x^3-21x^2 \phantom{00000000} \\[-3pt] \underline{-(-6x^3-18x^2)} \phantom{0000000} && -6x^2(x+3)=-6x^3-18x^2 \\[-3pt] 3x^2+9x \phantom{000} \\[-3pt] \end{array}

• And because the degree of our resulting polynomial is still higher than the degree of the divisor we can repeat once more.

\begin{array}{rll} 2x^3-6x^2+3x \phantom{000000} && \hbox{(Calculations - Steps 2 & 3)} \\[-3pt] x+3 \enclose{longdiv}{\cancel{2x^4}+\cancel{0x^3}\cancel{-21x^2}+\cancel{9x}+5}\kern-.2ex \\[-3pt] \underline{-(2x^4+6x^3)} \phantom{00000} \nabla \phantom{00000} \nabla \phantom{00} \nabla && 2x^3(x+3)=2x^4+6x^3 \\[-3pt] -6x^3-21x^2 \phantom{00000000} \\[-3pt] \underline{-(-6x^3-18x^2)} \phantom{0000000} && -6x^2(x+3)=-6x^3-18x^2 \\[-3pt] 3x^2+9x \phantom{000} \\[-3pt] \underline{-(3x^2+9x)} \phantom{00} && 3x(x+3)=3x^2+9x \\[-3pt] 5 \phantom{} \\[-3pt] \end{array}

• At last we are left with 5, which has a degree of 0. This is lower than the degree of the divisor (1) and as a result we can no longer divide any further. Because the resulting number is not 0, this means that we have a remainder. We can rewrite the final expression of our answer as follows:

\begin{array}{rl} 2x^3-6x^2+3x+\frac{5}{x+3} \end{array}

Notes:

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• Notice how the remainder is written as a fraction over the divisor.

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