Polynomial Long Division Summary

Polynomial Long Division

Learning Outcomes

  • Understand two methods for dividing polynomials, useful for finding zeros of non-factorable polynomials.
  • Recall the long division process, specifically for polynomials.

Reviewing Long Division

  • Long Division Overview: Rewriting the quotient as a polynomial of lower degree plus a remainder.
    • Expression format:
      [ p(x) = q(x)h(x) + r(x) ]
    • Where ( p(x) ) is the polynomial, ( q(x) ) is the divisor, ( h(x) ) is the quotient, and ( r(x) ) is the remainder.

Division Algorithm in Coordinate Systems

  • The division resembles ordinary fraction simplification, e.g., ( \frac{17}{3} = 5 + \frac{2}{3} ).
  • The long division algorithm includes:
    1. Divide the leading term of the dividend by the leading term of the divisor.
    2. Multiply the entire divisor by the result from step 1.
    3. Subtract this from the dividend.
    4. Repeat with the next term until the last term is processed.

Steps for Polynomial Long Division

divide 2x³ - 3x² + 4x + 5 by x + 2:

  1. Set Up Division Problem: Position x + 2 outside and 2x³ - 3x² + 4x + 5 inside.
  2. Determine First Term of Quotient:
    [ \frac{2x^3}{x} = 2x^2 ]
  3. Multiply and Write Below:
    [ 2x^2(x + 2) = 2x^3 + 4x^2 ]
  4. Subtract:
    [ (2x^3 - 3x^2 + 4x + 5) - (2x^3 + 4x^2) = -7x^2 + 4x + 5 ]
  5. Bring Down Next Term: Brings down the 5 to yield -7x² + 4x + 5.
  6. Repeat Steps: Continue finding the next term until reach the last term of the dividend.

Example Division of Polynomials

Example 1:

Divide: ( 5x^2 + 3x - 2 ) by ( x + 1 )

  • Set up Division Problem:
    • Calculate based on previous steps. Result: ( 5x - 2 )
  • No remainder: ( 5x² + 3x - 2 = (x + 1)(5x - 2) )

Example 2:

Divide: ( 6x^3 + 11x^2 - 31x + 15 ) by ( 3x - 2 )

  • Similar division method leads to remainder of 1.
  • Result can be expressed as revision of the division algorithm to illustrate:
    [ (3x - 2)(2x^2 + 5x - 7) + 1 = 6x^3 + 11x^2 - 31x + 15 ]

General Notes on Division Algorithm

  1. The Division Algorithm states that for a polynomial dividend ( f(x) ) and a non-zero divisor ( d(x) ), there exist unique polynomials ( q(x) ) (the quotient) and ( r(x) ) (the remainder).
  2. The properties of the remainder include:
    • Remainder should be either 0 or have a degree less than that of the divisor.
    • If ( r(x) = 0 ), it indicates that the divisor evenly divides the dividend.

Conclusion

  • Understanding polynomial long division is crucial for solving complex polynomial equations and finding roots that are non-factorable over integers.
  • The method reflects working with fractions and is backed by the Division Algorithm, providing a systematic approach for polynomial division.