Problem 16 - A Graphical Approach to College Algebra

Question

Factor each polynomial by grouping. $$2 m^{4}+6-a m^{4}-3 a$$

Step-by-Step Solution

Answer

The polynomial factors to \((2-a)(m^4 + 3)\).

Step 1 - Identify pair groups

The polynomial given is \(2m^4 + 6 - am^4 - 3a\). To factor by grouping, divide the polynomial into two pairs that can possibly be factored separately. Our pairs are \((2m^4 + 6)\) and \((-am^4 - 3a)\).

Step 2 - Factor out the greatest common factor in each group

For the first group \(2m^4 + 6\), the greatest common factor is 2. Factor 2 out: \[2(m^4 + 3)\].For the second group \(-am^4 - 3a\), the greatest common factor is \(-a\). Factoring \(-a\) out gives: \[-a(m^4 + 3)\].

Step 3 - Factor out the common binomial factor

Now that each group contains the same binomial \(m^4 + 3\), we can factor it out. Thus, the expression becomes:\[(2-a)(m^4 + 3)\]. The factorization is complete using the common binomial factor.

Key Concepts

Grouping Method

The process of polynomial factorization involves breaking down a polynomial into simpler expressions known as its factors. One efficient way to do this, particularly when dealing with polynomials with four terms, is the Grouping Method. In the Grouping Method, we divide the polynomial into groups or pairs of terms.

The aim is to organize these groups so that each can be factored further. In our original exercise, the polynomial is divided into two groups:

• Group 1: \(2m^4 + 6\)
• Group 2: \(-am^4 - 3a\)

By constructing these pairs, we increase our chances of recognizing common factors. This step is crucial because it allows us to factor out. Grouping provides a foundation that sets us up for the next crucial concept: finding the Greatest Common Factor (GCF).

By using the Grouping Method, we approach polynomial factorization systematically and often simplify our work considerably.

Greatest Common Factor

The Greatest Common Factor (GCF) is a key term in mathematics that refers to the highest factor shared by two or more numbers or terms. Identifying the GCF is a critical step in simplifying expressions, especially in polynomial factorization.

In the context of our exercise, after determining the groups, the next step is to factor out the GCF from each group:

• For Group 1 (\(2m^4 + 6\), the GCF is 2, resulting in \(2(m^4 + 3)\).
• For Group 2 (\(-am^4 - 3a\)), the GCF is \(-a\), leading to \(-a(m^4 + 3)\).

Recognizing and factoring out these greatest common factors simplifies each group to a form in which further operations, such as factoring out shared terms, can occur. Always remember, the GCF makes expressions easier to handle by reducing complexity.

After the GCF is applied, the expressions become more streamlined, paving the way for recognizing any common elements that arise between the groups.

Binomial Factor

In polynomial factorization, identifying and extracting a common binomial factor is often the final step, especially after using methods like grouping and finding the greatest common factor. A binomial factor is a simple two-term expression common to two separate algebraic expressions.

Within this polynomial, after factoring out the GCF from each group, we observe a shared binomial factor: \(m^4 + 3\). Both simplified groups contain the binomial \(m^4 + 3\):

• From Group 1: \(2(m^4 + 3)\)
• From Group 2: \(-a(m^4 + 3)\)

Since this binomial is present in both groups, it can be factored out completely, resulting in the final expression: \((2-a)(m^4 + 3)\).

Recognizing the binomial factor and factoring it out integrates the separate groups into one streamlined expression. The factorization is now complete, and the original polynomial is simplified into a product of two factors. This showcases how polynomials can often be reduced to their simplest forms, making them easier to work with in algebraic operations.