The POLY Menu is used to find all roots (or zeros) of any polynomial (up to 30th order).
Example 1: Find the roots of f(x) = x3 + 2x2 - x - 2
The values needed to solve for the roots of a polynomial are its order and the coefficients of each of its terms. The initial screen prompts for the order:
POLY
order=3
(we type three and press enter since the order of the polynomial is 3)
The display now prompts us to enter the coefficients of our polynomial by displaying:
a3x^3+...+a1x+a0=0 (representing a3x3
+ a2x2 +
a1x + a0 = 0) and then asking for each coefficient:
a3=1
a2=2
a1= -1
a0= -2
We type in the coefficents and press the down arrow or the enter key after each to move to the next one in the list. Missing terms in the polynomial they would have a coefficient of 0. Be sure to use the (-) key when entering negative coefficients. Now we press F5 to SOLVE for the roots. The display shows the 3 roots:
a3x^3+...+a1x+a0=0
x1= -2
x2= -1
x3=1
Graphing f(x) = x3 + 2x2 - x - 2 in the standard
window and then zooming in to (0,0) we verify these roots (see graph below).
The number of roots that the POLY routine finds is always the order of the polynomial. Some of these roots might be equal and some of them might be complex numbers.
Example 2: Find the roots of f(x) = x4 + 2x3 + 2x2 + 2x + 1
Enter its order and coefficients as before:
POLY
order=4
a4x^4+a3x^3+...+a1x+a0=0
a4=1
a3=2
a2=2
a1=2
a0=1
Now we press F5 to SOLVE for the roots.
The display shows the 4 roots (2 are complex and 2 are equal):
a4x^4+a3x^3+...+a1x+a0=0
x1=(0, 1) representing the complex root x1 = i
(after rounding 4.499...E-4 to 0)
x2=(0, -1) representing the complex root x2 = -i
(after rounding 4.499...E-4 to 0)
x3=(-1, 0) representing the real root x3= -1
(after rounding -1.00000031846 to -1)
x4=(-1, 0) representing the real root x4= -1 (multiplicity 2)
(after rounding -.999999681543 to -1)
Graphing f(x) = x4 + 2x3 +2x2 + 2x + 1 in the standard
window and then zooming in to (0,0) we verify there is only one real root
at -1 (see graph below).
