Problem 42 - Algebra 1: Concepts and Skills

Question

Factor the expression completely. \(x^{3}-3 x^{2}+x-3\)

Step-by-Step Solution

Answer

The fully factored form of the polynomial \(x^{3}-3 x^{2}+x-3\) is \((x-3)(x^{2}+1)\).

Step 1: Arrange the polynomial

The given polynomial is already arranged in the proper order: \(x^{3}-3 x^{2}+x-3\).

Step 2: Identify a common factor

There is no common factor for all the terms. We proceed to the next step.

Step 3: Break down into smaller polynomials

By looking closer, it can be seen that the polynomial can be written as two parts: \(x^{3}-3x^{2}\) and \(x-3\). Both can be factorized separately.

Step 4: Factorize the smaller polynomials

The first part has common factor x² that can be taken out, becomes \(x^{2}(x-3)\). The second part remains the same \(x-3\).

Step 5: Final factorization

Now, both parts have a common factor \(x-3\), hence the final factorization is \((x-3)(x^{2}+1)\).