Problem 69 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 2 x^{3}+4 x^{2}+x+2 $$

Step-by-Step Solution

Answer

The expression factors to \((x + 2)(2x^2 + 1)\).

Step 1: Group Terms

Group the terms of the polynomial to make it easier to factor. The expression is \(2x^3 + 4x^2 + x + 2\). Group it as: \((2x^3 + 4x^2) + (x + 2)\).

Step 2: Factor by Grouping

Factor out the greatest common factor from each group. From the first group, \(2x^3 + 4x^2\), factor out \(2x^2\) to get \(2x^2(x + 2)\). From the second group, \(x + 2\), there's no need to factor anything further as it is already simplified.

Step 3: Factor Out Common Binomial

Notice that both groups now have \((x + 2)\) as a common factor. Factor \((x + 2)\) from the entire expression to obtain \((x + 2)(2x^2 + 1)\).

Step 4: Confirm Completeness of Factoring

Check if the polynomial \(2x^2 + 1\) can be factored further. Since it doesn’t have a factorable pattern or real roots, it can't be factored further. The complete factorization of the expression is \((x + 2)(2x^2 + 1)\).

Key Concepts

Grouping Method

The grouping method is a technique used to factor polynomials that have four or more terms. This method involves rearranging and grouping terms in a polynomial to make the expression easier to factor. In our example polynomial expression, \(2x^3 + 4x^2 + x + 2\), we start by grouping the terms into two pairs: \((2x^3 + 4x^2)\) and \((x + 2)\). Notice how each group is set up to reveal a common factor.

When using the grouping method:

• Look for terms that can be paired with others to share a common factor.
• Re-group the polynomial neatly, observing any signs that come with each term to avoid mistakes.
• Factor out the common factor from each group.

Employing this method not only organizes the polynomial but also sets the stage for more straightforward factorization techniques to be applied thereafter.

Greatest Common Factor

Finding the greatest common factor (GCF) of terms involves identifying the largest factor that is common among the terms in a polynomial. In the expression \(2x^3 + 4x^2\), both terms share a numerical component of \(2\) and a variable component of \(x^2\). Thus, the GCF is \(2x^2\). Extracting this GCF is the first step in simplifying polynomial groups.

Steps to find the GCF:

• Break down each component of the terms into their prime factors (for numbers) and smallest degree (for variables).
• Select the lowest power of each common factor shared by both terms.
• Multiply these factors together to get the GCF, which is then factored out.

By factoring out the GCF from each group, the polynomial becomes easier to manipulate and further factor.

Binomial Factoring

The binomial factoring technique focuses on identifying and extracting a common binomial factor from a polynomial expression. For the polynomial \(2x^3 + 4x^2 + x + 2\), after grouping and identifying the GCF, we arrive at \(2x^2(x + 2) + 1(x + 2)\). Here, \((x + 2)\) is a common binomial factor.

Key aspects of binomial factoring include:

• Identifying repeating binomial expressions within grouped terms.
• Factoring these out to simplify the expression further.

In this particular case, once \((x + 2)\) is factored out, you're left with \((x+2)(2x^2 + 1)\). This approach highlights how recognizing patterns in polynomial expressions leads to streamlined solutions, offering a straightforward way to factor efficiently.