Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{3}-3 x^{2}-x+3 $$

Short answer

(x + 1)(x - 1)(x - 3)

Step-by-step solution

- Group the terms

First, group the polynomial into two pairs: \( (x^3 - 3x^2) + (-x + 3) \).

- Factor out the greatest common factor (GCF) from each pair

Factor out the GCF from each group: \( x^2(x - 3) - 1(x - 3) \).

- Factor by grouping

Notice that \(x - 3\) is a common factor. Factor \(x - 3\) out: \( (x^2 - 1)(x - 3) \).

- Factor the difference of squares

\( x^2 - 1 \) is a difference of squares: \( (x + 1)(x - 1) \). Therefore, the polynomial is completely factored as: \( (x + 1)(x - 1)(x - 3) \).

Key concepts

greatest common factor

When factoring polynomials, the first step is often to find the Greatest Common Factor (GCF). The GCF is the largest factor that can evenly divide all the terms of the polynomial.
To find the GCF, follow these steps:

• Write down each term of the polynomial.
• Find the common factors of the coefficients (numbers in front of the variables).
• Identify the smallest power of the common variables.

For instance, if we have the polynomial: \(x^{3} - 3x^{2} - x + 3\), we group the terms into pairs: \( (x^{3} - 3x^{2}) + (-x + 3) \). For the first group \( (x^{3} - 3x^{2}) \), the GCF is \(x^{2}\). For the second group \( (-x + 3) \), the GCF is \(-1\).
By factoring the GCF from each pair, we simplify the polynomial.

factoring by grouping

Factoring by grouping involves rearranging and grouping terms in such a way that we can factor out common factors in pairs.
Here's how to do it:

• Group the terms of the polynomial into pairs.
• Factor out the GCF from each group.
• If done correctly, a common binomial factor should emerge from the groups.

Taking our example: \(x^{3} - 3x^{2} - x + 3\), we group to get \((x^{3} - 3x^{2}) + (-x + 3)\).
We factor out the GCF from each group: \(x^{2}(x - 3) - 1(x - 3)\).
Now, \(x - 3\) is a common binomial factor in both groups. We factor out \(x - 3\) to get: \((x - 3)(x^{2} - 1)\).
This simplifies our polynomial significantly.

difference of squares

The difference of squares is a special form of factoring that occurs when you have a binomial in the form: \(a^{2}-b^{2}\). This can be factored as:\((a + b)(a - b)\).

• Identify the terms that are squared.
• Apply the difference of squares formula.

For our example, we have: \(x^{2} - 1\). Recognizing that \(1 = 1^{2}\), we see: \(x^{2} - 1^{2}\).
Applying the difference of squares, we get: \((x + 1)(x - 1)\).
So, our polynomial factored becomes: \((x + 1)(x - 1)(x - 3)\). This method often greatly simplifies complex polynomials.

polynomial factorization

Polynomial factorization is the process of breaking down a polynomial into simpler, non-factorable polynomials, called factors. These factors, when multiplied together, give back the original polynomial.
It follows steps such as:

• Identifying the GCF.
• Grouping terms and factoring by grouping.
• Applying special factoring formulas like the difference of squares.

For our exercise polynomial: \(x^{3} - 3x^{2} - x + 3\), we combine these steps.
We grouped terms, factored out the GCF, and used the difference of squares:
- Grouped: \((x^{3} - 3x^{2}) + (-x + 3)\)
- GCF Factored: \(x^{2}(x - 3) - 1(x - 3)\)
- Common Factor: \((x - 3)(x^{2} - 1)\)
- Difference of Squares: \((x - 3)(x + 1)(x - 1)\)
These steps help in breaking down the polynomial into factors that are easier to work with and understand.