Problem 20 - 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. $$ 2 n^{2}+12 n-5 m n-30 m $$

Step-by-Step Solution

Answer

Answer: \((n + 6)(2n - 5m)\)

Step 1: Group the terms

The given polynomial is: $$ 2n^2 + 12n - 5mn - 30m $$ Group the terms in pairs: $$ (2n^2 + 12n) + (-5mn - 30m) $$

Step 2: Factor each group

Now, we will factor each group: The first group: \((2n^2 + 12n)\) Find the greatest common factor of \(2n^2\) and \(12n\), which is \(2n\). Then, factor \(2n\) out of each term: $$ 2n(n + 6) $$ The second group: \((-5mn - 30m)\) Find the greatest common factor of \(-5mn\) and \(-30m\), which is \(-5m\). Then, factor \(-5m\) out of each term: $$ -5m(n + 6) $$

Step 3: Check for common factors

Now that we've factored each group, let's see if there are any common factors: The factored groups are: $$ 2n(n + 6) - 5m(n + 6) $$ We can see that the common factor here is \((n + 6)\).

Step 4: Factor the common factor out

Factor the common factor \((n + 6)\) out of the expression: $$ (n + 6)(2n - 5m) $$ So, the factored polynomial using the grouping method is: $$ (n + 6)(2n - 5m) $$

Key Concepts

Grouping Method

The grouping method is a strategy used to factor polynomials by rearranging and grouping terms into sets that have a common factor. This technique is particularly useful for polynomials that do not have a single common factor across all terms. It allows us to factor by identifying common factors within smaller groups of terms. In essence, we split the polynomial into pairs or sets, and then factor each pair individually. The goal is to identify a common factor between the groups so that the polynomial can be expressed as a product of two binomials.

Greatest Common Factor

The greatest common factor, or GCF, is the largest factor that divides each term of a set of terms. Finding the GCF is crucial when factoring polynomials, as it helps simplify the expression and makes it easier to factor. To find the GCF of algebraic terms, you need to consider both the coefficient (numerical part) and the variable part:

• For coefficients, identify the largest number that divides all the given numbers.
• For variables, consider the smallest power for each common variable across the terms.

This ensures that you extract the largest possible factor from each term, simplifying the work of factoring.

Factoring by Grouping

Factoring by grouping involves using the dichotomy of the polynomial into smaller, manageable expressions. Take the polynomial in question: \[2n^2 + 12n - 5mn - 30m\].We begin by creating groups: \[(2n^2 + 12n) + (-5mn - 30m)\].
This makes it easier to find and factor out the GCF from each group:

• First group: GCF of \(2n^2\) and \(12n\) is \(2n\). Thus, it factors into \(2n(n + 6)\).
• Second group: GCF of \(-5mn\) and \(-30m\) is \(-5m\). Resulting in \(-5m(n + 6)\).

At this point, both groups share a common expression, \((n + 6)\). Factoring this out gives us:\[(n + 6)(2n - 5m)\].
This method demonstrates how grouping facilitates recognizing common factors, leading to a simplified form of the polynomial.

Polynomial Expressions

Polynomial expressions are mathematical statements with variables and coefficients, involving operations of addition, subtraction, and non-negative integer exponents. Understanding the structure of polynomials is fundamental to applying the grouping method successfully. In polynomials:

• Terms can include constants, variables, or both. They are separated by plus or minus signs.
• The degree of a polynomial is determined by the highest degree of its terms.
• Each term's degree is the sum of the exponents of the variables in that term.

Given a polynomial like \[2n^2 + 12n - 5mn - 30m\],
analyzing its terms and attempting to reconfigure them using methods like grouping pinpoints potential simplifications and factorizations. This is crucial as efficient manipulation of polynomial expressions allows more complex problems to be tackled with confidence.