Problem 119 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(2 x^{3}-x^{2}+6 x-3\)

Step-by-Step Solution

Answer

The completely factored expression is \((x^2 + 3)(2x - 1)\).

Step 1: Group the terms

First, group the terms in pairs to find common factors within those pairs. Group the expression as follows: \[(2x^3 - x^2) + (6x - 3)\]

Step 2: Factor out the GCF from each pair

Identify and factor out the greatest common factor (GCF) from each of the grouped pairs. For the first group \(2x^3 - x^2\), the GCF is \(x^2\):\[x^2(2x - 1)\]For the second group \(6x - 3\), the GCF is \(3\):\[3(2x - 1)\]

Step 3: Factor by grouping

Notice both terms now have a common binomial factor \(2x - 1\). Factor out this common binomial:\[(x^2 + 3)(2x - 1)\]

Step 4: Verify factors are completely factored

Check if the factors \(x^2 + 3\) and \(2x - 1\) can be factored further. \(x^2 + 3\) is non-factorable over the real numbers since it's a sum of squares. \(2x - 1\) is already a linear factor. Thus, the complete factorization is \[(x^2 + 3)(2x - 1)\]

Key Concepts

Group factoring

When faced with complex polynomials, one effective method to simplify them is group factoring. This process involves rearranging the terms and grouping them into pairs or sets that share common factors.
For instance, consider the polynomial given: \(2x^3 - x^2 + 6x - 3\). To begin with group factoring, we first regroup the terms: \((2x^3 - x^2) + (6x - 3)\).

• No change in the value of the expression occurs by grouping the terms.
• This regrouping is a foundational step that helps us identify commonalities within smaller sections of the polynomial.

After grouping, we can inspect each group separately to find common factors. Group factoring is particularly useful when dealing with polynomials that do not initially present an obvious common factor across all terms.

Greatest Common Factor (GCF)

The next approach in polynomial factoring involves identifying and extracting the greatest common factor (GCF) from each group you created. The GCF of a set of terms is the biggest expression that can divide each of them without leaving a remainder.

• For the first group \(2x^3 - x^2\), the GCF is \(x^2\), since both terms have at least an \(x^2\) to factor out.
• For the second group \(6x - 3\), the GCF is \(3\), as both terms share a factor of 3.

Factoring the GCF from each group will leave you with:\[x^2(2x - 1)\] from the first group and\[3(2x - 1)\]from the second group.
This step efficiently simplifies the polynomial and prepares it for further factoring by revealing common structure and factors across groups.

Binomial factorization

After identifying and factoring out the GCF in each group, the key to advancing in polynomial factorization by grouping is recognizing common binomial factors.
In our example, once you factor the GCF from each group, a common binomial factor, \((2x - 1)\), can be observed:

• \(x^2(2x - 1)\)
• \(3(2x - 1)\)

This commonality allows the expression to be written in the form \((x^2 + 3)(2x - 1)\) by factoring out \((2x - 1)\), similar to how you might factor out a common number in arithmetic.
This method leverages the repeated presence of binomials to further simplify and factorize polynomial expressions.

Complete factorization

Once a polynomial has been simplified through methods like group factoring and the extraction of the GCF, the final step is to ensure all factors are as simplified as possible. This process is known as complete factorization.
Examine the factors you obtained:\((x^2 + 3)\) and \((2x - 1)\).

• \(x^2 + 3\) is a sum of squares and cannot be further factored over the real numbers, as it does not present any real roots.
• \(2x - 1\) is already in its simplest linear form.

Thus, the expression \((x^2 + 3)(2x - 1)\) is deemed completely factored since no further simplification is possible using real number factorization. Complete factorization assures that the expression is broken down into its simplest, most basic building blocks that multiply to give the original polynomial.