Using Polynomial Long Division Without Remainders

Description/Explanation/Highlights

Video Description

This video explains how to use polynomial long division to solve division of two polynomials that does not result in a remainder.

Steps and Key Points to Remember

To use long division to solve a polynomial division problem, follow these steps:

  1. To divide two polynomials such as \((x^2+3x-40)\div(x-5)\) using long division, first rewrite the problem as follows:  \(\hspace{20 pt}x-5)\overline{\hspace{5 pt}x^2+3x-40}\)
  2. 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 \(x\) by \(x\) would equal the \(x^2\) in the dividend. Write it above the x term (3x) in the dividend as follows: \(\\\hspace{80 pt}x\\\hspace{20 pt}x-5)\overline{\hspace{5 pt}x^2+3x-40}\)
  3. Next, multiply the \(x\) on top by the entire divisor by distributing \(x\) to both terms and write the answer below the dividend. \(\\\hspace{80 pt}x\\\hspace{20 pt}x-5)\overline{\hspace{5 pt}x^2+3x-40}\\\hspace {51 pt}x^2-5x\)
  4. Draw a line underneath and subtract. Be sure to apply the subtraction sign to both terms. It may be helpful to use parenthesis. Notice how subtracting \(-5x\) had the effect of adding \(5x\) to \(3x\) and subtracting \(x^2\) eliminated \(x^2\) by making it equal to zero.\(\\\hspace{80 pt}x\\\hspace{20 pt}x-5)\overline{\hspace{5 pt}x^2+3x-40}\\\hspace {35 pt}-(\underline{x^2-5x)\hspace{20 pt}}\\\hspace{74 pt}8x\)
  5. Bring down the next term and put it beside the result from subtracting. In the above example, bring down \(-40\) and put it beside \(8x\)  \(\\\hspace{80 pt}x\\\hspace{20 pt}x-5)\overline{\hspace{5 pt}x^2+3x-40}\\\hspace {35 pt}-(\underline{x^2-5x)\hspace{20 pt}}\\\hspace{74 pt}8x-40\)
  6. 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. In our example that number is 8 since 8 times x will equal 8x. Place the 8 above -40 (the constant). \(\\\hspace{80 pt}x+8\\\hspace{20 pt}x-5)\overline{\hspace{5 pt}x^2+3x-40}\\\hspace {35 pt}-(\underline{x^2-5x)\hspace{20 pt}}\\\hspace{74 pt}8x-40\)
  7. Multiply the term by the entire divisor as before and subtract as before. If the result of the subtraction is 0, there is no remainder and the problem is finished. \(\\\hspace{80 pt}x+8\\\hspace{20 pt}x-5)\overline{\hspace{5 pt}x^2+3x-40}\\\hspace {35 pt}-(\underline{x^2-5x)\hspace{20 pt}}\\\hspace{74 pt}8x-40\\\hspace{58 pt}-(\underline{8x-40})\\\hspace{102 pt}0\)

Here are some key points to keep in mind when dividing polynomials 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.
  • 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 there is a missing variable add it in to the polynomial with a coefficient of zero, for example: write \(2x^3-4x+5\) as \(2x^3+0x^2-4x+5\) when writing the dividend or divisor.
  • If the final result at the bottom is zero, there is no remainder.
  • If there is a remainder, it needs to be written as a fraction. Please see my video titled Using Polynomial Long Division Resulting in Remainders to learn more about what to do with remainders.

Video Highlights

  • 00:00 Introduction
  • 00:21 \((x^2+3x-40)\div(x-5)\) example with steps to solve
  • 03:26 \((x^3-2x^2-9x+18)\div(x^2-5x+6)\) example of a 4-term polynomial divided by a trinomial
  • 06:26 Conclusion
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