Question

What is the complete factorization of the polynomial below?

x3 - 3x2 + x - 3

Answer 1

The complete factorization of the polynomial x³ - 3x² + x - 3 is (x - 3)(x² + 3)(x + 1).

Step-by-step explanation:

The complete factorization of the polynomial x³ - 3x² + x - 3 can be found by using the Rational Root Theorem and synthetic division. By trying out the possible rational roots using the Rational Root Theorem, we find that one of the roots is 3. So, we divide the polynomial by (x - 3) to get the quadratic factor and the remaining linear factor.

(x - 3) is a factor, so we divide x³ - 3x² + x - 3 by (x - 3) using synthetic division:
3|1-31-3|3030|1010
The result is x² + 3.

The remaining linear factor is 1x + 1 = x + 1.

Therefore, the complete factorization of the polynomial x³ - 3x² + x - 3 is (x - 3)(x² + 3)(x + 1).

Answer 2

The complete factorization of the given polynomial is:

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

Step-by-step explanation:

Factorization of a polynomial--

It means that the polynomial could be expressed as the product of distinct factors containing the rational roots of the polynomial.

We are given a polynomial expression as:

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

Now, it could be factorized as follows:

`x^3-3x^2+x-3=x^2(x-3)+1(x-3)\\\\i.e.\\\\x^3-3x^2+x-3=(x^2+1)(x-3)`

Now, we know that the expression:

`x^2+1` do not have a rational root.

Hence, the complete factorization is:

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