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10.5 – Dividing Polynomials

Key Terms

  • Dividend – The number to be divided (numerator).
  • Divisor – The number being divided by (denominator).
  • Quotient – The result of a division problem (answer).
  • Remainder – A number left over after dividing whole numbers.
    • The letter r is sometimes used to represent a remainder.



  • Constant terms have a degree of zero
    • Ex. 4=4x^{0}
    • Ex. -36=-36x^{0}

  • For long division, use DMS, and perform these steps over and over until you are done!
    1. Divide
    2. Multiply
    3. Subtract


What is Division?
  • When you divide, you separate a total amount into equal groups to find the number of groups or to find the number in each group.
  • You cannot divide by zero because there is no way to separate a total into zero groups or into groups with zero items.
    • Ex. If I have 33 oreos and 14 students, I could divide the oreos by the number of students to see how many cookies each student would receive.  There would be a remainder of 5, as each student would receive 2 cookies each.
    • Ex. If I have 58 students who want to play Capture the Flag, and we only want 2 teams, how many students could be on each team?  You’d want to divide the students into 2 groups!  There would be 29 students on each team!


Parts of a Division Problem
Alg1B 10.5 Parts of Division


Steps for Dividing Polynomials Using Long Division
  1. Make sure the divisor and dividend are both written in descending order.
    1. Ex. Change 11x^{2}+9+4x^{3}\;\; into \;\; 4x^{3}+11x^{2}+9
  2. Fill in any missing terms by writing a term with a coefficient of zero.
    1. 4x^{3}+11x^{2}+0x+9
  3. Identify any values for x that would make the problem impossible to solve.
    1. If you divide by x+3, x cannot be -3 because it would make the divisor zero (which is undefined).
  4. DMS: Divide, Multiply, Subtract
    1. Divide: ask yourself, “how many times does the 1st term in the divisor go into the 1st term in the dividend?”
      1. Place the answer in the quotient
    2. Multiply that term of the quotient by all terms in the divisor
    3. Subtract the entire remainder (you may have to distribute the negative sign)
    4. Repeat until  you have a remainder of a constant term or zero

  • Step-by-Step
  • Another Example (no remainder: zero)

Alg1B 10.5 Ex with Zeros


When to Stop Dividing
  • Ask yourself:
    • Is the degree (exponent) of the remainder less than the degree (exponent) of the divisor?
      • If no, I should keep dividing.
      • If yes, I should stop dividing.
        • Example: For (3x^{2}+4x-5)\div (x+2), the quotient is (3x-2) remainder -1.
        • Remainder -1 has a degree of zero.
        • Zero is less than x^{1}, so stop dividing!


Check Your Work
  • Division and multiplication are inverse operations.
  • Use multiplication to check division.
    • divisor • quotient = dividend
    • Example (using the same problem in the example above)

Alg1B 10.5 Check Work

  • If the answer has a remainder, you need addition to check the answer.
    • (divisor • quotient) + remainder = dividend


  • Ex 1. Use long division to find the quotient: (4x^{2}-7x-2)\div (x-2)

Alg1B 10.5 Q3

Answer: 4x+1

  • Ex 2. Use long division to find the quotient: (16x^{3}+4x^{2}-144)\div (4x-8)

Alg1B 10.5 Q6

  • Ex 3. In a polynomial division problem, the dividend does not have an x^{4} term.
    When the problem is written in long division form, what should be the coefficient of the x^{4} term in the dividend?

    • Answer: 0
  • Ex 4. For what value of x would the quotient (2x^{2}+3x-4)\div (x+4) not make sense?
    Answer: -4 because -4 + 4 = 0, and you cannot have zero in the denominator. It would be undefined.

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