TI-86 Tutorial

Factoring any Polynomial

The POLY Menu can be used to find all roots (or zeros) of any polynomial (up to 30th order). All we need to do is take advantage of the connection between the roots of a polynomial and its factors:

If r is a root of a polynomial then (x - r) is a factor.

Example 1: Factor the polynomial x3 + 2x2 - x - 2

Use POLY to find its roots:

The display shows the 3 roots:

x1= -2
x2= -1
x3=1

The root -2 gives the factor (x + 2).
The root -1 gives the factor (x + 1).
and the root 1 gives the factor (x - 1).

So, x3 + 2x2 - x - 2 = (x + 2)(x + 1)(x - 1)


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.

If r is a root with multiplicity k then (x - r)k is a factor of the polynomial.

If c = A + Bi is a complex root then the conjugate of c, A - Bi, is also. So (x - (A + Bi) and (x - (A - Bi)) are both factors of the polynomial. We usually want the factors to have real coefficients. If we multiply these two linear factors together we get the quadratic factor that has real coefficients ( x2 - 2Ax + (A2 + B2) ).

Example 2: Factor x4 + 2x3 + 2x2 + 2x + 1

Using POLY we find 4 roots (the first 2 are complex and the last 2 are equal):

x1=(0, 1) representing the complex root x1 = i
x2=(0, -1) representing the complex root x2 = -i
x3=(-1, 0) representing the real root x3= -1
x4=(-1, 0) representing the real root x4= -1

The real root of -1 (multiplicity 2) gives the factors (x + 1)2

The complex roots i (so, A=0 and B=1) and its conjugate -i give the factors (x - i)(x + i) = (x2 + 1).

So, x4 + 2x3 + 2x2 + 2x + 1 = (x + 1)2(x2 + 1)