![]() Graphing a Piecewise Defined Function with Three Pieces In the Y 2= field, repeat Steps 2, 3, and 4 to input the second piece of the piecewise defined function and its domain. Press ENTER, input the rest of the inequality to express the domain of the first function, and then input a right parenthesis. Scroll down to the desired inequality symbol for the domain of the first function. ![]() Press 2nd, then MATH to open the TEST menu. In the Y 1= field, input a left parenthesis, the first function, a right parenthesis, a division bar, a left parenthesis, and then X. Input the two pieces of the piecewise defined function separately by entering the functions enclosed in parentheses as numerators and the domains of the functions enclosed in parentheses as denominators, as follows. Graphing a Piecewise Defined Function with Two Pieces X 3 ≈ 0.52932 x_3 \approx 0.52932 x 3 ≈ 0.To evaluate the function at a given value of x while the graph is displayed, press TRACE and then input the given value of x. If b 2 = 3 a c b^2 = 3ac b 2 = 3 a c, then the polynomial has a triple root: If Δ = 0 \Delta = 0 Δ = 0, then the polynomial has three real roots, and at least two of them are equal. If Δ < 0 \Delta< 0 Δ < 0, then the polynomial has one real root and two non-real complex conjugate roots. If Δ > 0 \Delta > 0 Δ > 0, then the polynomial has three distinct real roots. The sign of Δ \Delta Δ provides us with some knowledge about the roots of our polynomial. In particular, the sign of Q 3 + R 2 Q^3 + R^2 Q 3 + R 2 is opposite to that of the discriminant. If you don't succeed, use the cubic equation formula, which is not the most user-friendly method in mathematics but always yields the correct result! You may also try plotting the polynomial and guessing its root from the graph. Should the polynomial have a rational root, this method will find it. ![]() If your polynomial has rational coefficients, try performing the rational root test (or use the rational zeros calculator to do it for you). To perform the division, you may want to use the method described in the synthetic division calculator.īut how to find the initial root? Well, there are no easy and 100% successful recipes. Then you need to divide your cubic polynomial by x − q x - q x − q to arrive at a quadratic polynomial. If you are somehow able to determine one root, then finding the other two poses no problem since your task reduces to solving a quadratic equation, which you can do either by factoring (as in the factoring trinomials calculator) or by using the quadratic formula. Fortunately, there's Omni's cubic equation calculator, which can find the roots of any cubic equation in no time! It's definitely more complicated than in the case of quadratic trinomials, where we have the well-known quadratic formula. In general, finding the roots of cubic equations may be challenging. In the latter case, they are a pair of conjugate numbers, i.e., their real parts are equal, and their imaginary parts have opposite signs. The other two roots might be real or complex. Has a root 0 0 0 with multiplicity three.Ī cubic equation always has at least one real root. Some of these roots, however, may be equal. It follows from the fundamental theorem of algebra that every cubic equation has exactly three complex roots. A root of a cubic equation is every argument x x x that satisfies this cubic equation.Ģ 3 − 8 = 8 − 8 = 0 2^3 - 8 = 8 - 8 = 0 2 3 − 8 = 8 − 8 = 0.
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