Integration Using Long Division (College Board AP® Calculus AB)

Integration using long division

How do I divide polynomials?

• It is possible to use polynomial long division to simplify a rational function like

  • This can make the function much easier to integrate

• Polynomial division works just like the long division method used for regular numbers:

• The answer to a polynomial division is built up term by term

  • working downwards in powers of the variable (usually x)

• E.g. dividing by

• (the thing being divided) is known as the dividend

  • and (the thing we're dividing by) is known as the divisor

• Start by dealing with the highest power term in the dividend ()

  • Compare the highest power term in the divisor ()

  • so

    • puton top of the division line

    • and subtract from the dividend

•  Now deal with the highest power remaining in the expression on the bottom line ()

  • Compare the highest power term in the divisor ()

  • so

    • addon top of the division line

    • and subtract from the bottom line

• Now deal with the highest power remaining in the expression on the bottom line ()

  • Compare the highest power term in the divisor ()

  • so

    • addon top of the division line

    • and subtract from the bottom line

• The 3 'left over' at the bottom is the remainder

  • Therefore

  • In that new form, the function would be very easy to integrate