Problem 59 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ x^{3}+2 x^{2}+5 x+10 $$

Step-by-Step Solution

Answer

The polynomial factors as \((x + 2)(x^2 + 5)\).

Step 1: Group the Terms

Group the polynomial terms into two pairs: \( (x^3 + 2x^2) + (5x + 10) \).

Step 2: Factor Out the Greatest Common Factor (GCF) from Each Group

For the first group \( (x^3 + 2x^2) \), the GCF is \( x^2 \), so factor it out: \( x^2(x + 2) \).For the second group \( (5x + 10) \), the GCF is \( 5 \), so factor it out: \( 5(x + 2) \).

Step 3: Factor Out the Common Binomial Factor

Now that both groups contain the common factor \( (x + 2) \), factor \( (x + 2) \) out of each expression: \( x^2(x + 2) + 5(x + 2) = (x + 2)(x^2 + 5) \).

Key Concepts

Grouping Method

The grouping method is a helpful technique when you're dealing with polynomials that don’t look easy to factor at first glance. This method is particularly useful for polynomials with four terms, as it involves strategically grouping terms to simplify the expression.

Here's how the grouping method works:

• Identify logical pairs of terms within the polynomial.
• Group these terms in a way that allows for a common factor to be extracted from each group.
• Factor out the greatest common factor from each identified pair.

The idea is to rearrange or group the polynomial terms in such a manner that factoring becomes feasible, as demonstrated in factoring, say, a polynomial like: \[ (x^3 + 2x^2) + (5x + 10) \]. Here, terms are grouped into two separate pairs, making the extraction of a common factor possible. When the polynomial, as grouped, can’t be factored further, one should try a different arrangement until it can be simplified, or determine that it's not factorable by this method.

Greatest Common Factor

The greatest common factor (GCF) represents the largest factor that divides each term of the polynomial. Identifying and extracting the GCF is a crucial step in simplifying expressions, especially in the context of the grouping method.

To find the GCF:

• Look at each term within a group to determine the highest number or algebraic term they all share.
• Factor this term out, leading to a simpler polynomial expression.

For example, in the polynomial group \(x^3 + 2x^2\), the GCF is \(x^2\). Extracting \(x^2\) simplifies this group to \(x^2(x + 2)\). Similarly, in \(5x + 10\), the GCF is \(5\), so it factors to \(5(x + 2)\).

Factoring out the GCF not only simplifies the polynomial but also uncovers common factors across different terms, making the polynomial easier to manipulate and solve.

Binomial Factor

A binomial factor is simply a two-term expression of the form \(a + b\) or \(a - b\). In polynomial factorization, the goal is often to identify a binomial factor that is common across different terms.

Once the initial grouping and factoring of greatest common factors are done, the next step involves looking for any binomial expressions that are common to each group. In our exercise, after factoring out the GCF from each pair, we have \((x + 2)\) as the common binomial factor across both groups: \(x^2(x + 2)\) and \(5(x + 2)\).

By extracting this common binomial factor \((x + 2)\), what remains in each group is combined into a new binomial: \((x + 2)(x^2 + 5)\). This step drastically simplifies the expression and completes the factorization of the polynomial.

Four-Term Polynomial

Four-term polynomials are expressions with exactly four terms, making them a good candidate for the grouping method. These polynomials often require strategies beyond the basic factoring techniques, given their complexity and number of terms.

For example, the polynomial given in the original exercise, \(x^3 + 2x^2 + 5x + 10\), has four distinct terms. Rather than attempting straightforward factoring, which can be complex with multiple terms, we utilize the grouping method to simplify the task into more manageable parts.

The steps are straightforward:

• First, identify and group the terms into pairs that make sense to work with.
• Next, each group undergoes factoring by the greatest common factor.
• Finally, identify and factor out common binomial expressions.

Using such methods, a seemingly difficult expression becomes manageable, allowing us to find the factors and simplify the expression effectively.