How to Solve Polynomial Exponents to the Third
- 1). Graph the third-degree polynomial. If the graphed curve crosses the x-axis three times, or crosses it one time and just touches it one time, all three roots are real. If it just touches the x-axis once, there is a multiple root --- two identical roots. If the place where the curve crosses or touches the x-axis is p, one of the binomial factors will be x - p. Graphing calculators are hard to read, so all of these candidates should be checked by dividing them into the third-degree equation. If there isn't a remainder, the binomial is a factor.
- 2). Find the candidate binomial factors for the third-degree equation by looking at all combinations of the first and last number in the third-degree equation. For example, in 2x^3 - 6x^2 - 2x + 6, the first number is 2, which has factors of 1 and 2, and the last number is 6, which has factors 1, 2, 3 and 6. The candidates are x - 1, x + 1, x - 2, x + 2, x - 3, x + 3, x - 6, x + 6, 2x - 1, 2x + 1, 2x - 2, 2x + 2, 2x - 3, 2x + 3, 2x - 6 and 2x + 6.
- 3). Try dividing all the candidate binomial factors to find that only x - 1, x + 1 and 2x - 6 = x + 3 divide the third-degree polynomial without leaving a remainder. Therefore, this leaves 2x^3 - 6x^2 - 2x + 6 = (x - 1)(x + 1)(2x - 6). Setting each binomial to zero and solving produces the roots -1, 1 and 3.
Source...