Find the roots of a cubic equation

Enter a cubic's four coefficients, read its roots, and check them by factorisation.

3 steps

Suppose an algebra exercise asks for the roots of:

x^{3} - 6 x^{2} + 11 x - 6 = 0

The Polynomial equation mode checks the roots after you identify the four coefficients. Write the equation as $a x^{3} + b x^{2} + c x + d = 0$ : here $a = 1$ , $b = - 6$ , $c = 11$ , and $d = - 6$ . The coefficient boxes take numbers, not the letter $x$ .

Find the roots on CalcES

Below each step is the screen you should see after pressing.

Next to each step is the screen you should see after pressing.

Step 1

Open the mode list, choose Polynomial equation, then degree 3. The four boxes $a, b, c, d$ run from the cubic term to the constant.

Press:
Step 2

Enter $1$ , $- 6$ , and $11$ in the first three boxes. Press the equals key after each coefficient to advance.

Press:
Step 3

Enter $- 6$ in box $d$ and press the equals key to solve. Read all the roots shown; their display order is not significant.

Press:

Check the answer

The roots are ${1, 2, 3}$ . Factorisation confirms why:

x^{3} - 6 x^{2} + 11 x - 6 = (x - 1) (x - 2) (x - 3)

Each root makes one factor zero. If your course asks you to show working, use the calculator to check your answer and present the factorisation or your own algebraic method as well.

If the roots differ, check that the equation equals zero and that the four coefficients were entered in descending-power order. Unlike degree 2, the degree-3 result here does not display a turning point.

Find the roots on CalcES

Step 1

Step 2

Step 3

Check the answer