What is Polynomial Long Division?

Polynomial long division is a method used to divide one polynomial by another polynomial of a lower degree, similar to the long division process with numbers.

Steps to Perform Polynomial Long Division

  1. Set Up the Division
    Write the dividend (the polynomial you’re dividing) and the divisor (the polynomial you are dividing by) in long division format. For example, let’s divide $2x^3 + 3x^2 – x + 5$ by $x – 1$

  1. Divide the Leading Terms
    Divide the leading term of the dividend by the leading term of the divisor. In our example, divide $2x^3$ by $x$ to get $2x^2$. Write $2x^2$ above the division bar.

  1. Multiply and Subtract
    Multiply $2x^2$ by the entire divisor $(x – 1)$, giving $2x^3 – 2x^2$. Subtract this from the original dividend, resulting in a new polynomial: $5x^2 – x + 5$

  1. Repeat the Process
    Repeat the steps with the new polynomial. Divide the leading term of $5x^2$ by $x$ to get $5x$. Multiply $5x$ by $x – 1$ to get $5x^2 – 5x$. Subtract this from $5x^2 – x + 5$ to get $4x + 5$

  1. Continue Until Remainder is Smaller Degree
    Continue this process until the degree of the remaining polynomial is less than the degree of the divisor. Divide $4x$ by $x$ to get $4$. Multiply $4$ by $x – 1$ to get $4x – 4$. Subtract this from $4x + 5$ to get $9$. The final quotient is $2x^2 + 5x + 4$ with a remainder of $9$

Example

Let’s go through an example step by step:

  1. Setup: Divide $2x^3 + 3x^2 – x + 5$ by $x – 1$
  2. First Division: $2x^3 / x = 2x^2$
  3. Multiply and Subtract: $2x^2 times (x – 1) = 2x^3 – 2x^2$. Subtract to get $5x^2 – x + 5$
  4. Second Division: $5x^2 / x = 5x$
  5. Multiply and Subtract: $5x times (x – 1) = 5x^2 – 5x$. Subtract to get $4x + 5$
  6. Third Division: $4x / x = 4$
  7. Multiply and Subtract: $4 times (x – 1) = 4x – 4$. Subtract to get $9$
  8. Result: Quotient is $2x^2 + 5x + 4$, remainder is $9$

Conclusion

Polynomial long division helps in simplifying complex polynomial expressions and finding quotients and remainders. This method is essential in algebra for solving polynomial equations and simplifying rational expressions.

Citations

  1. 1. Khan Academy – Polynomial Long Division
  2. 2. Purplemath – Polynomial Long Division
  3. 3. Math is Fun – Polynomial Long Division

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