factor x^3 - 2x^2 + x - 2

Factor:

x^3 - 2x^2 + x - 2

Note the pattern of coefficients:
+1-2 +1-2
which allows you to factor as follows:
=x²(x-2) + 1(x-2)
=(x²+1)(x-2)

thank you so much! :)

You're welcome!

To factor the expression x^3 - 2x^2 + x - 2, we can try to use the rational root theorem to find any possible rational roots.

The rational root theorem states that if a polynomial has a rational root of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p is among the possible factors of the constant term and q is among the possible factors of the leading coefficient.

In this case, the constant term is -2 and the leading coefficient is 1. The factors of -2 are ±1 and ±2, and the factors of 1 are ±1.

Therefore, the possible rational roots are ±1, ±2.

To check if any of these possible roots are actually roots of the polynomial, we can substitute each one into the polynomial and see if it equals to zero.

For x = 1, we have (1)^3 - 2(1)^2 + (1) - 2 = 1 - 2 + 1 - 2 = -2.

For x = -1, we have (-1)^3 - 2(-1)^2 + (-1) - 2 = -1 - 2 - 1 - 2 = -6.

For x = 2, we have (2)^3 - 2(2)^2 + (2) - 2 = 8 - 8 + 2 - 2 = 0.

Therefore, x = 2 is a root of the polynomial.

Using synthetic division, we can divide the polynomial by (x - 2) to find the other factor. The factors obtained from the synthetic division will be the coefficients of the quadratic factor.

Performing the synthetic division, we have:

2 | 1 - 2 1 - 2
---------------------
| 2 0 2
---------------------
1 0 1 0

The result of the synthetic division gives us the coefficients of the quadratic factor, which are 1, 0, and 1. Rewriting this, we have:

x^3 - 2x^2 + x - 2 = (x - 2)(x^2 + 1)

Therefore, the factored form of the expression x^3 - 2x^2 + x - 2 is (x - 2)(x^2 + 1).