Algebra - Cubics

The Factor Theorem and Cubic Equations

The Factor Theorem establishes that a polynomial $P(x)$ has a factor $(x - k)$ if and only if $P(k) = 0$. Conversely, if $P(k) = 0$, then $x = k$ is a root of the polynomial equation $P(x) = 0$. Exam cubics usually have at least one integer root. If a cubic with integer coefficients has an integer root, that root must be a factor of the constant term, so test those factors (positive and negative) first.

To solve a cubic equation like $2x^3 - 3x^2 - 17x + 30 = 0$, begin by testing integer factors of $30$, such as $\pm 1, \pm 2, \pm 3$. Evaluating $P(-3)$ gives $2(-27) - 3(9) - 17(-3) + 30 = -54 - 27 + 51 + 30 = 0$. Because $P(-3) = 0$, $(x + 3)$ is a factor.

Next, divide the cubic polynomial by $(x + 3)$ using algebraic long division. This division yields the quadratic quotient $2x^2 - 9x + 10$. Factoring the quadratic gives $(2x - 5)(x - 2)$. Setting each factor to zero provides the complete set of roots: $x = -3$, $x = 2$, and $x = \frac{5}{2}$.