Problem 16 - Elementary Algebra

Question

For the following problems, use the grouping method to factor the polynomials. Some polynomials may not. be factorable using the grouping method. $$ 3 n^{2}+6 n+9 m^{3}+12 m $$

Step-by-Step Solution

Answer

Answer: The factored form is \(3n(n + 2) + 3m(3m^2 + 4)\).

Step 1: Identify the common factors in each pair of terms

First, we'll focus on the first two terms: \(3n^2\) and \(6n\). We can see that they both have a common factor 3n. Now if we look at the next set of terms: \(9m^3\) and \(12m\), they have a common factor of 3m.

Step 2: Factor the common factors

Now we'll factor the common factors from each pair of terms to simplify the polynomial. $$(3n^2 + 6n) + (9m^3 + 12m)$$ $$= 3n(n + 2) + 3m(3m^2 + 4)$$

Step 3: Check if further factoring is possible

In the factored polynomial \(3n(n + 2) + 3m(3m^2 + 4)\), we cannot factor anything else using the grouping method as there is no common factor between the two grouped terms. Thus, the final factored form of the given polynomial using the grouping method is: $$ 3n(n + 2) + 3m(3m^2 + 4) $$

Key Concepts

Grouping Method

The grouping method is a helpful tool for factoring polynomials, especially when dealing with four or more terms. It involves organizing the terms into pairs to find common factors within each group.
This technique can simplify the polynomial and make it easier to factor further, if necessary. The main goal is to rewrite the polynomial so that it becomes a set of simpler expressions that can be further manipulated or factored.
To apply the grouping method, follow these steps:

• Identify and group terms into pairs that could have common factors.
• Factor out the common factors from each pair, transforming the polynomial into a sum of products.
• Examine whether it is possible to combine the resulting expressions into one single factored expression by identifying additional common factors.

The key to success with the grouping method lies in correctly identifying pairs of terms to group together. Not all polynomials can be factored by this method, but it frequently works well for those divisible into pairs.

Common Factors

In polynomials, common factors are terms or values that divide exactly into each term of a polynomial. Recognizing and factoring out these common factors can simplify expressions significantly.
When using the grouping method, identifying common factors within each set of grouped terms is crucial. For example, in the original exercise, common factors were identified from the pairs:
- First pair: 3n was a common factor for the terms \(3n^2\) and \(6n\)- Second pair: 3m was a common factor for \(9m^3\) and \(12m\)
Finding these common factors requires observation and sometimes a bit of trial and error, but the reward is a simpler expression that can often be factored further.
Keep in mind, sometimes after factoring out the most obvious common factors, the polynomial may still have hidden common factors that can further simplify it.

Polynomials

Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.
They are important in many areas of algebra, serving as the foundation for many algebraic operations and techniques.
Each term within a polynomial is typically composed of a coefficient (a constant number) and one or more variables raised to a power.

• For example, in the term \(3n^2\), 3 is the coefficient and \(n\) is the variable.

Polynomials can be classified by the number of terms they contain, such as binomials (two terms), trinomials (three terms), or more complex polynomials with higher numbers of terms.
Understanding polynomials and their properties is critical in algebra, particularly when learning how to perform operations like factoring, which often involves recognizing patterns and simplifying expressions.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations or understand relationships between quantities.
It forms the basis for more advanced studies in mathematics and sciences. Central to algebra is the concept of expressing calculations in a generalized form using variables and constants.
One of the key skills in algebra is factoring, which involves breaking down complex expressions into simpler pieces built from the product of factors.
Learning to factor expressions such as polynomials means recognizing patterns and applying techniques like the grouping method to simplify calculations and solve equations more easily.
Engaging with algebraic concepts such as polynomials helps develop problem-solving skills and algebraic thinking, enabling students to apply these techniques to various real-world contexts.