Identify the leading coefficient (a) and the constant term (d) in the equation. The Rational Zero principle states that any zeros of the equation will be a rational number using factors of the leading coefficient as the numerator and factors of the constant term as the denominator.
Factor the leading coefficient and constant term. For example, for the equation 2x^3 + 2x^2 + 3x -6, the leading coefficient is 2 with factors 1, -1, 2 and -2. The constant is -6, so the factors are 1, -1, 2, -2, 3, -3, 6 and -6. Thus, the total set of factors is 1, -1, 2, -2, 3, -3, 6 and -6.
Test each factor for a zero output in the function. Mark which values produce a zero. These values are the x-intercepts of the equation.
Factor the equation into its lowest forms using polynomial factoring methods. The factoring process attempts to reduce the equation into a form with no exponents. This will depend on the type of equation and the leading coefficients; e.g., x^3 + 2x^2 -- x -- 2 = (x -- 1)(x + 1)(x + 2).
Use the factored equation to determine the multiplicity or number of each x-intercept. An x-intercept can be identified by taking each factored grouping and setting it equal to zero. For example, the equation x(x -- 2)(x + 3) would have x-intercepts of 0, 2 and -3 with multiplicities of one. The equation (x -- 1)(x -- 1)(x + 1) would have a multiplicity two x-intercept 1 and a multiplicity one x-intercept -1.
Determine the transitivity of each x-intercept. If an x-intercept has odd multiplicity, it is considered transitive and is a point where the equation will cross the x-axis. Even multiplicity determines an intransitive x-intercept, where the equation will not cross the x-axis.
Locate the leading coefficient (a). Negative coefficients create a graph line under the x-axis beyond the largest zero. If the coefficient is positive, it will be above the x-axis.
On graph paper or on a set of axes, plot the x-intercepts.
Enter more values between the highest and lowest zeros into the equation to plot additional points. More points will create a smoother graph line.
Draw a line roughly fitting the plotted points.