Using Polynomial Long Division Resulting in Remainders

Description/Explanation/Highlights

Video Description

This video explains how to use polynomial long division to solve division of two polynomials that results in a remainder.

Steps and Key Points to Remember

To use long division to solve a polynomial division problem and write its remainder as a fraction, follow these steps:

  1. If you don’t know how to do basic polynomial long division without remainders, I would suggest first referring to the page: Using Polynomial Long Division Without Remainders and watching the associated video to learn the basics of long division with polynomials.
  2. To divide two polynomials such as \((2x^4+5x^2-2x+3)\div(x+2)\) using long division, first rewrite the problem as follows making sure to add the missing \(x^3\) with a coefficient of 0:  \(\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\)
  3. Compare the first term in the divisor (outside of the box) to the first term in the dividend (inside the box) and determine what you would multiply the first divisor term by to get the first dividend term. In the above example, multiplying \(2x^3\) by \(x\) would equal the \(2x^4\) term in the dividend. Write it above the \(x^3\) term (\(0x^3\)) in the dividend as follows:\(\\\hspace{78 pt}2x^3\\\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\)
  4. Next, multiply the \(2x^3\) on top by the entire divisor by distributing \(2x^3\) to both terms and write the result below the dividend. \(\\\hspace{78 pt}2x^3\\\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\\\hspace {51 pt}2x^4+4x^3\)
  5. Draw a line underneath and subtract. Be sure to apply the subtraction sign to both terms. It may be helpful to use parenthesis. \(\\\hspace{78 pt}2x^3\\\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\\\hspace {35 pt}-(\underline{2x^4+4x^3)\hspace{20 pt}}\\\hspace{65 pt}-4x^3\)
  6. Bring down the next term in the dividend and put it beside the result from subtracting. \(\\\hspace{78 pt}2x^3\\\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\\\hspace {35 pt}-(\underline{2x^4+4x^3)\hspace{20 pt}}\\\hspace{65 pt}-4x^3+5x^2\)
  7. Compare the first term in the divisor to the first term in the new polynomial as before to see what the divisor needs to be multiplied by to equal the first term of the polynomial. Place it above the corresponding term in the dividend. \(\\\hspace{78 pt}2x^3-4x^2\\\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\\\hspace {35 pt}-(\underline{2x^4+4x^3)\hspace{20 pt}}\\\hspace{65 pt}-4x^3+5x^2\)
  8. Multiply the term by the entire divisor as before and subtract as before.  \(\\\hspace{78 pt}2x^3-4x^2\\\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\\\hspace {35 pt}-(\underline{2x^4+4x^3)\hspace{20 pt}}\\\hspace{65 pt}-4x^3+5x^2\\\hspace {53 pt}-(\underline{-4x^3-8x^2)\hspace{20 pt}}\\\hspace{99 pt}13x^2\)
  9. Bring down the next term from the dividend and repeat the process of comparing, multiplying and subtracting. \(\\\hspace{78 pt}2x^3-4x^2+13x\\\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\\\hspace {35 pt}-(\underline{2x^4+4x^3)\hspace{20 pt}}\\\hspace{65 pt}-4x^3+5x^2\\\hspace {53 pt}-(\underline{-4x^3-8x^2)\hspace{20 pt}}\\\hspace{99 pt}13x^2-2x\\\hspace{82 pt}-(\underline{13x^2+26x})\\\hspace{117 pt}-28x\)
  10. Repeat the process once more by bringing down the last term in the dividend and following the same steps. \(\\\hspace{78 pt}2x^3-4x^2+13x-28\\\hspace{20 pt}x+2)\overline{\hspace{5 pt}2x^4+0x^3+5x^2-2x+3}\\\hspace {35 pt}-(\underline{2x^4+4x^3)\hspace{20 pt}}\\\hspace{65 pt}-4x^3+5x^2\\\hspace {53 pt}-(\underline{-4x^3-8x^2)\hspace{20 pt}}\\\hspace{99 pt}13x^2-2x\\\hspace{82 pt}-(\underline{13x^2+26x})\\\hspace{117 pt}-28x+3\\\hspace{105 pt}-(\underline{-28x-56})\\\hspace{156 pt}59\)
  11. Since all terms in the dividend have been brought down, whatever is left is the remainder. It is generally written in the final answer as a fraction with the remainder over the divisor. In the above example the final answer would be written: \(2x^3-4x^2+13x-28+\frac{59}{x+2}\) Note: Although this remainder does not have a variable, variables can and often do occur in the remainder.

Here are some key points to keep in mind when dividing polynomials that will result in remainders when using long division.

  • Long division can be used to divide any polynomial by another.
  • Write the division problem as you would a long division problem using numbers with the dividend in the “box” and the divisor (what you are dividing by) on the outside.
  • If there is a missing variable add it in to the polynomial with a coefficient of zero, for example: write \(2x^4+5x^2-2x+3\) as \(2x^4+0x^3+5x^2-2x+3\) when writing the dividend or divisor.
  • At each step, compare the first term of the polynomial to the first term of the divisor to determine what to multiply the divisor by to “eliminate” the first term of the polynomial.
  • If after bringing down all terms, comparing, multiplying, and subtracting, the final result at the bottom is anything besides zero, there is a remainder.
  • Remainders are generally written as a fraction added at the end of the solution polynomial with the remainder as the numerator and the divisor as the denominator. For example a remainder of \(5x-1\) would be written as \(\frac{5x-1}{x^2-4x+5}\) if the divisor in the problem is \(x^2-4x+5\).

Video Highlights

  • 00:00 Introduction
  • 00:12 \((x^3-3x^2+6x+4)\div(x^2-4x+5)\) example that results in a remainder, along with steps to solve
  • 03:45 \((2x^4+5x^2-2x+3)(x+2)^{-1}\) division example written with a negative exponent, a missing \(x^3\) term, and resulting in a remainder
  • 09:15 Conclusion
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