Division of Polynomials

Long Division and Synthetic Division

Polynomials

A polynomial is an expression that may contain constants, variables and exponents, that can be combined using addition, subtraction, multiplication and division, but a polynomial will never have division by a variable, it will never have an infinite number of terms, and the exponents of it's variables must be 0, 1, 2, 3, 4, ... etc.

Dividing Polynomials:

Methods of dividing polynomials, long division and synthetic division, will be the topic of this newsletter, as we explore the process by which one can use long division or synthetic division to divide two polynomials.

Long Division

Steps to complete long division of polynomials:

  1. Set up the division problem as you would any long division problem comprised completely of constants, with the original polynomial under the long division symbol and the polynomial by which the original is divided outside the long division symbol. Make sure the polynomial is written in descending order of exponents, and be sure to use a zero as a placeholder for exponential values not represented in the polynomial.
  2. Find the value by which the first term in the divisor is multiplied by to equal the first term in the dividend, and write this value above the second term of the dividend.
  3. Multiply the value you just placed over the second term of the dividend (the quotient) by the entire divisor and place the product beneath the first and second term of the dividend.
  4. Multiply the term you just placed under the first two terms of the dividend by -1 and subtract the two values from the first two values in the dividend.
  5. By doing the steps above, the first two terms you've subtracted should cancel out and you should be left with a remainder from the subtraction of the second terms. Bring down the next term of the dividend so that it is in line with this remainder and then repeat steps 2-5 until there are no more terms in the dividend to drag down and you are left with a remainder (which could also be zero).
Long Division of Polynomials
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Synthetic Division

Steps to complete synthetic division of polynomials:

  1. Set the denominator (the polynomial being divided from the original polynomial) equal to zero and solve for x. Place this value of x in a box to the left of where the rest of your work will go.
  2. Making sure the terms in your numerator (original polynomial from which the other is divided) are in descending order according to their exponent values. Then, take the coefficient in front of each term in the numerator (if the term is just x or x^2, then your coefficient is 1) and line them up in the same order to the right of the box you made in step 1. Draw a line under all of the coefficients and the box leaving a little room in between.
  3. Drawing an arrow, bring down the first coefficient from the list down below the line you drew in step 2.
  4. Multiply this term by the value in the box and then place the product below the next term of the list of coefficients, right above the line.
  5. Add the next term of the list of coefficients by the number just placed below it and place the sum below both terms and beneath the line.
  6. Repeat step 4 and 5 for each term in the list of coefficients until there are no more coefficients to be used and you reach a remainder (which must be written as a fraction) or a zero.
  7. The new values you've placed below the line represent the coefficients of the quotient polynomial, and all that is required to reach the answer is simply plugging in these coefficients in to a polynomial starting with one exponential value less than the original polynomial (for example, if the dividend polynomial started with x to the third degree, the highest degree on an x value of the quotient would be 2).
Synthetic Division of Polynomials
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Summary

Polynomials may contain many or few terms (which can be constants, variables, and/or exponents) and they can be divided by other polynomials. Two different methods of dividing polynomials are long division and synthetic division, and the process by which one may divide by either of these methods has been detailed in this newsletter.

Upcoming Dates

  • Quiz on December 7th or 8th
  • Test on December 10th