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5.3 – Synthetic Division

Key Terms

  • Descending Order – In order from highest power to lowest (exponents), from left to right.
    • This is the standard way to write a polynomial.
  • Dividend – The number to be divided.
  • Divisor – The number being divided by.
  • Quotient – The result of a division problem.
  • Remainder – A number left over after dividing whole numbers. The letter r is sometimes used to represent a remainder.
  • Synthetic Division – A process for dividing polynomials.
    • This method is known as the “shortcut” for dividing polynomials because it requires fewer calculations than does long division.
    • It ONLY works when the divisor is in linear form: (x\pm c).



  • Parts of Division Problems

Alg2B 5.3 Div Poly Labels


  • Integer Long Division with Fractional Remainder

Alg2B 5.3 Div w Remainder Fraction


  • Polynomial Long Division (No Remainder)
    • (x^2+3x+2) \div(x+1)
    • x≠-1 because -1+1=0, and you cannot divide by zero (undefined)




  • Synthetic Division
    • Always write polynomials in descending order (highest degree exponent first).
      • Ex. 3x+2x^4+5-6x^5-x^3+3x^2 should be rewritten in this order: -6x^5+2x^4-x^3+3x^2+3x+5
    • If any terms are missing, insert them with a coefficient of zero.
      • Ex. 2x^2+5 should be written 2x^2+0x+5, where the zero acts as a placeholder for the x-term.
    • Synthetic Division is a shortcut for polynomial division
    • Below are the STEPS for Solving with Synthetic Division

Alg2B 5.3 Synthetic Rules

      • Take a look at step 2, above.  The “a” of (x – a) is the divisor .
        • Ex. If (x – 2) is your divisor, you would use “2” as your divisor term on the outside.
        • Ex. If (x + 4), change it to (x – (-4)), and use “-4” as your divisor term on the outside.
        • So, you should notice that you always use the opposite of whatever divisor you are given:
          • Ex. (x – 6) would be +6
          • Ex. (x + 3) would be -3

Alg2B 5.3 Synthetic Div Setup


  • Ex. (x^2-2x+5) \div(x+4)
    • First, rewrite the divisor as (x-(-4)) so that you know your “a” is -4.

Alg2B 5.3 Labels



  • Remainders
    • Synthetic division may have fractional remainders, just like integer division

Alg2B 5.3 Div w Remainder


  • Ex 1: Synthetic Division Setup

Alg2B 5.3 Example01


  • Ex 2. Solving with Synthetic Division

Alg2B 5.3 Example02



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