Problem 18 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 8 m^{2} n^{3}-24 m n^{4} $$

Step-by-Step Solution

Answer

The expression factors to \(8mn^3(m - 3n)\).

Step 1: Identify Common Factors

Look at both terms in the expression \(8m^2n^3\) and \(24mn^4\) to identify any common factors. Both terms have the factors of \(m\) and \(n\), and the coefficients 8 and 24 share a common factor of 8.

Step 2: Factor Out the Greatest Common Factor (GCF)

The greatest common factor for the coefficients is 8, and both terms contain at least one \(m\) and three \(n\)'s. Thus, the GCF for the entire expression is \(8mn^3\).Factor out \(8mn^3\) from each term: \[8m^2n^3-24mn^4 = 8mn^3(m - 3n)\]

Step 3: Simplify the Expression

After factoring out the GCF, we are left with the expression in parentheses: \(m - 3n\). The factored expression is:\[8mn^3(m - 3n)\]

Step 4: Check for Further Factoring

Analyze the remaining expression \(m - 3n\) to determine if it can be factored further. Since there are no additional common factors or any recognizable patterns (like a trinomial), the expression is fully factored.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a crucial concept when factoring polynomials. It involves finding the largest number or expression that is a factor of each term in a polynomial. When dealing with multiple terms, like in algebraic expressions, the goal is to identify and extract the GCF, simplifying the problem into more manageable parts.
For example, in the problem given \(8m^{2}n^{3}-24mn^{4}\), we look at each term's numeric coefficient and variable components separately.

• Identify the largest number that divides evenly into all the coefficients. For the numbers 8 and 24, the largest factor is 8.
• For the variables, identify common variables with the smallest exponent. Here, both terms share the variable \(m\) and \(n\). The smallest power of each is \(m\) and \(n^3\).

Therefore, the GCF of the entire expression is \(8mn^3\). Factoring out the GCF simplifies the expression to \(8mn^3(m-3n)\).
Using the GCF streamlines the factoring process and often reduces errors in algebraic manipulations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations, forming a meaningful mathematical relationship. Understanding these components is essential for efficiently solving algebraic problems, like the one presented.
Algebraic expressions can vary from simple linear forms such as \(x+2\), to more complex constructs like \(8m^2n^3-24mn^4\). These expressions can include terms; each term is a product of coefficients and variables raised to powers. In the example provided, there are two terms, each a distinct combination of coefficients (8 and 24) and variable parts (

• The first term \(8m^2n^3\) includes a coefficient 8 and variables \(m\) and \(n\), raised to the powers of 2 and 3, respectively.
• The second term \(24mn^4\) has a coefficient 24, with the variables \(m\) and \(n\), raised to the powers of 1 and 4.

Recognizing these structures within algebraic expressions aids significantly in identifying methods for simplification and factoring.

Factoring Techniques

The process of factoring involves breaking down an expression into a product of simpler terms, allowing for easier manipulation or solving of equations. Factoring techniques can vary, but the most basic and commonly used method involves identifying and extracting the greatest common factor.In our example \(8m^2n^3-24mn^4\), a technique involves these steps:

• First, identify the GCF from all terms, which includes both numeric coefficients and shared variable components. For our specific expression, the GCF we identified and factored out was \(8mn^3\).
• Write down the factored form, reflecting the extracted GCF and the reduced residual terms. This gives the result \(8mn^3(m-3n)\).
• Finally, verify the simplified form of the algebraic expression to ensure it cannot be factored further. In the case of the expression \(m-3n\), no further factorization is possible.

These methods not only simplify expressions but also provide insights when solving equations that include polynomials. Mastery of factoring techniques empowers students to tackle increasingly complex algebraic problems with confidence.