At this point it is necessary to expand each of the polynomials, multiply the results by the coefficients, and combine like terms. Do this and you will get a polynomial of the form:

This equation has five possible roots, since the highest powered term is of degree 5. So now you need to look up the general solution to the quintic equation and plug in the coefficients and calculate the five possible roots. There is just one problem, though. There is no general solution to the quintic equation. Never was and never will be (that was proven by a mathematician named Abel about 200 years ago). There are general solutions up to the quartic (highest degree is 4), but those will not help here. This problem cannot be solved with a simple mathematical formula.

There is a practical way around this problem called interpolation, and that is how such problems are solved. It is exactly like playing the children’s game “battleship”. You

guess at an answer, and if it is right, great. If it is wrong, guess again. In this case, you make a guess at the appropriate discount rate and solve for the NPV. If you guess 6%, the NPV would be around $21. Since the NPV is positive, you had not discounted the cash flows by a large enough value, so try again. This time if you guess 10%, you would get an NPV of about -$21, so you had discounted too much and need to try a lower value. At a discount rate of 8%, you get an NPV of $0, and this is the IRR.

But there is still a difficulty. The original problem was quintic and had five possible roots; 8% is only one of them. There are still four more roots out there somewhere. As it turns out, for this problem all five roots are simultaneous, that is they are all 8%. The reason this is so is because of the pattern of the cash flows. On the time line, the first cash flow at t=0 is negative, and then the sign of the cash flow changes and all the