How to Solve Polynomial Exponents to the Third

Polynomials with exponents to the third degree always have a real-valued root.

Third-degree polynomials are terms where the largest exponent is three. The degree equals the number of roots. Complex roots always come in pairs, so expect at least one real-valued root for a degree-three polynomial. If there is only one real-valued root, the third-degree equation can be factored into a binomial and a trinomial. Solving the trinomial by using the quadratic equation will produce the two complex-valued roots.

Things You'll Need

• A graphing calculator

Instructions

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.

Tips & Warnings

• You can save a lot of time if you use the information from the graph to find which of the binomial factor candidates to try first. If you find them right away, you can save a lot of time by not testing the rest of the candidates.

• The information from the graph can be confusing or hard to read. It is necessary to check by trying to divide by the factor candidates.

References

• WTAMU: Polynomials
• MathIsFun: Polynomials
• TheMathPage: Roots of Polynomials of Degree Less Than Two

Resources

• PurpleMath: Solving Polynomials