Problem 60 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ x^{3}+4 x^{2}+3 x+12 $$

Step-by-Step Solution

Answer

The polynomial factors as \((x^{2}+3)(x+4)\).

Step 1: Group Terms

To factor by grouping, divide the polynomial into two groups. For the expression \( x^{3} + 4x^{2} + 3x + 12 \), group the terms as follows: \( (x^{3} + 4x^{2}) + (3x + 12) \).

Step 2: Factor Each Group

Now, factor out the greatest common factor (GCF) from each grouped term: - For \( x^{3} + 4x^{2} \), the GCF is \( x^{2} \). Factoring gives \( x^{2}(x + 4) \).- For \( 3x + 12 \), the GCF is \( 3 \). Factoring gives \( 3(x + 4) \).

Step 3: Combine Common Factors

Notice that both groups have a common binomial factor \( (x + 4) \). Factor this out: \( x^{2}(x + 4) + 3(x + 4) = (x^{2} + 3)(x + 4) \).

Step 4: Write Final Factored Form

The polynomial \( x^{3} + 4x^{2} + 3x + 12 \) is fully factored and the factored form is \( (x^{2} + 3)(x + 4) \).

Key Concepts

Factoring by Grouping

Factoring by grouping is a strategic method used to factor polynomials, particularly useful when dealing with four-term polynomials. The goal is to pair the terms into groups that can each be factored separately before combining them to simplify the expression. This method relies on recognizing patterns and ensuring groups share similar characteristics that allow for simplification.
Here's how to approach factoring by grouping:

• Identify appropriate pairs of terms: Start by dividing the polynomial into two pairs. For example, in the polynomial \(x^{3} + 4x^{2} + 3x + 12\), you may group it as \((x^{3} + 4x^{2}) + (3x + 12)\).
• Factor each group: In each identified group, factor out the greatest common factor (GCF).
• Combine like terms: After factoring each group, look for any common binomial factors, then factor them out to simplify further.

This method is a blend of careful observation and systematic approach, leading to neat simplifications that reveal the inherent symmetry of polynomial expressions.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is an essential concept when it comes to factoring, as it is the largest factor that divides all terms in a group without leaving a remainder.
To find and factor the GCF:

• List out factors: Identify the smallest factors and their powers for each term in a group.
• Select the highest shared factor: Choose the factor that appears in all terms with the lowest power.
• Factor it out: Extract this factor to simplify the group, as seen in the grouping \(x^{3} + 4x^{2}\), where \(x^{2}\) is the GCF, resulting in \(x^{2}(x + 4)\).

Understanding the GCF is crucial as it facilitates breaking down complex expressions into manageable ones, providing clarity when dealing with polynomial expressions.

Binomial Factors

Binomial factors are expressions consisting of two terms that appear repeatedly across different groups within a polynomial expression. Identifying and extracting binomial factors is a key step in simplifying and factoring polynomials by grouping.
In the example \((x^{3} + 4x^{2}) + (3x + 12)\), after factoring out the GCF from each group, we obtain \(x^{2}(x + 4) + 3(x + 4)\). Here, \((x + 4)\) is a common binomial factor.
Steps to recognize and factor out binomial factors:

• After extracting GCF, observe if a binomial appears in each factored group.
• If found, factor out this common term as seen here, resulting in \((x^{2} + 3)(x + 4)\).

Recognizing binomial factors not only simplifies the expression but also often results in simpler solutions or polynomial factorization, facilitating easier computation or further manipulation.