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

Integration using long division

How do I divide polynomials?

• It is possible to use polynomial long division to simplify a rational function like $\frac{x^3+6x^2-9x-11}{x-2}$

  • 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 $x^3+6x^2-9x-11$ by $x-2$

• $x^3+6x^2-9x-11$(the thing being divided) is known as the dividend

  • and $x-2$(the thing we're dividing by) is known as the divisor

• Start by dealing with the highest power term in the dividend ($x^3$)

  • Compare the highest power term in the divisor ($x$)

  • $x^3÷x=x^2$ so

    • put$x^2$on top of the division line

    • and subtract $x^2·\left(x-2\right)=x^3-2x^2$ from the dividend

•  Now deal with the highest power remaining in the expression on the bottom line ($8x^2$)

  • Compare the highest power term in the divisor ($x$)

  • $8x^2÷x=8x$ so

    • add$8x$on top of the division line

    • and subtract $8x·\left(x-2\right)=8x^2-16x$ from the bottom line

• Now deal with the highest power remaining in the expression on the bottom line ($7x$)

  • Compare the highest power term in the divisor ($x$)

  • $7x÷x=7$ so

    • add$7$on top of the division line

    • and subtract $7·\left(x-2\right)=7x-14$ from the bottom line

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

  • Therefore $\frac{x^3+6x^2-9x-11}{x-2}=x^2+8x+7+\frac{3}{x-2}$

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