Polynomial Division (DP IB Analysis & Approaches (AA)): Revision Note

Polynomial Division

What is polynomial division?

• Polynomial division is the process of dividing two polynomials

  • This is usually only useful when the degree of the denominator is less than or equal to the degree of the numerator

• To do this we use an algorithm similar to that used for division of integers

• To divide the polynomial  by the polynomial   where k ≤ n

  • STEP 1
    Divide the leading term of the polynomial P(x) by the leading term of the divisor D(x) : 

  • STEP 2
    Multiply the divisor by this term:

  • STEP 3
    Subtract this from the original polynomial P(x) to cancel out the leading term:

  • Repeat steps 1 – 3 using the new polynomial R(x) in place of P(x) until the subtraction results in an expression for R(x) with degree less than the divisor

    • The quotient Q(x) is the sum of the terms you multiplied the divisor by:

    • The remainder R(x) is the polynomial after the final subtraction

Division by linear functions

• If P(x) has degree n and is divided by a linear function (ax + b) then

  •  where 

    • ax + b is the divisor (degree 1)

    • Q(x) is the quotient (degree n – 1)

    • R is the remainder (degree 0)

  • Note that 

Division by quadratic functions

• If P(x) has degree n and is divided by a quadratic function (ax² + bx + c) then

  •   where

    • ax² + bx + c is the divisor (degree 2)

    • Q(x) is the quotient (degree n – 2)

    • ex + f is the remainder (degree less than 2)

  • The remainder will be linear (degree 1) if e ≠ 0, and constant (degree 0) if e = 0

  • Note that

Division by polynomials of degree k ≤ n

• If P(x) has degree n and is divided by a polynomial D(x) with degree k ≤ n
  

  •   where

    • D(x) is the divisor (degree k)

    • Q(x) is the quotient (degree n – k)

    • R(x) is the remainder (degree less than k)

  • Note that 

Are there other methods for dividing polynomials?

• Synthetic division is a faster and shorter way of setting out a division when dividing by a linear term of the form

  • To divide  by :

    • Set 

    • Calculate 

    • Continue this iterative process 

    • The quotient is  and the remainder is

• You can also find quotients and remainders by comparing coefficients

  • Given a polynomial 

  • And a divisor 

  • Write and 

  • Write 

    • Expand the right-hand side

    • Equate the coefficients

    • Solve to find the unknowns q’s & r’s