Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{4}-x^{3}+x-1 $$

Short answer

(x - 1)(x + 1)(x^{2} - x + 1)

Step-by-step solution

Group the terms

Group the polynomial into two pairs: \[ x^{4} - x^{3} \text{ and } x - 1 \]

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

Factor out the common factor from each group: \[ x^{3}(x - 1) + 1(x - 1) \]

Factor by grouping

Now factor out the common binomial factor \(x - 1\) from the grouped terms: \[ (x - 1)(x^{3} + 1) \]

Factor the resulting binomial

Factor the sum of cubes \(x^{3} + 1\) using the formula \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\), where \(a = x\) and \(b = 1\): \[ x^{3} + 1 = (x + 1)(x^{2} - x + 1) \]

Combine all factors

Combine all the factors obtained: \[ (x - 1)(x + 1)(x^{2} - x + 1) \]

Key concepts

Grouping Terms

Grouping terms is a crucial first step in factoring certain types of polynomials. When you group, you essentially rearrange the polynomial into smaller, more manageable segments. In the exercise, we started by splitting the original polynomial, \(x^{4}-x^{3}+x-1\), into two groups: \(x^{4}-x^{3}\) and \(x-1\). This approach lets you focus on smaller parts of the polynomial, making it easier to identify common factors in each group.

Greatest Common Factor

After grouping, the next step is to factor out the Greatest Common Factor (GCF) from each group. The GCF is the highest degree term that is common across all terms in each group. For example, in \(x^{4} - x^{3}\), the GCF is \(x^{3}\), and in \(x - 1\), the GCF is 1. By factoring these out, we get \ (x^{3}(x - 1) + 1(x - 1)) \. This step simplifies each group and makes it easier to spot further factoring opportunities.

Sum of Cubes

Factoring polynomials using the sum of cubes formula is another method often used after initial grouping and factoring out the GCF. In the step by step solution, after grouping and factoring out the GCF, we had the expression \((x - 1)(x^{3} + 1)\). To factor \(x^{3} + 1\), we use the sum of cubes formula: \a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\. Here, \(a = x\) and \(b = 1\), giving us \ (x^{3} + 1 = (x + 1)(x^{2} - x + 1))\.

Factoring by Grouping

Factoring by grouping is where you combine the groups formed in earlier steps to fully factor the polynomial. In this example, we started with the grouped terms \(x^{3}(x - 1) + 1(x - 1)\). Noticing that both groups had the common binomial \(x - 1\), we factored this out, resulting in \ (x - 1)(x^{3} + 1)\. Factoring completely involves addressing each of the resulting components.

Binomial Factor

Finally, using the binomial factors identified throughout the process allows us to combine terms into the final factored form. We originally found \(x - 1\) as a binomial factor in step three, then factored \(x^{3} + 1\) in step four. Combining these gives us the final result: \ (x-1)(x+1)(x^{2}-x+1) \. Using binomials is efficient and simplifies the process, making it easier to see the structure of the factored polynomial.