Problem 14 - 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. $$ 36 a k-8 a h-27 b k+6 b h $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial is 8a(4k - h) - 3b(9k - 2h).

Step 1: Group the terms of the polynomial.

First, we will group the terms of the polynomial into pairs as follows: $$ (36ak - 8ah) + (-27bk + 6bh) $$

Step 2: Factor the greatest common factor from each pair.

Now, we will factor out the greatest common factor from each pair of terms: $$ 8a(4k - h) - 3b(9k - 2h) $$

Step 3: Check for any common factor between the two groups.

Observe that there is no common factor between the two groups (other than 1), so the polynomial cannot be factored further using the grouping method. The final factored form of the given polynomial using the grouping method is: $$ 8a(4k - h) - 3b(9k - 2h) $$

Key Concepts

Grouping Method

The grouping method is a popular technique for factoring polynomials, especially when dealing with four-term polynomials. It involves rearranging and grouping terms into smaller chunks that can be factored individually. By breaking down the polynomial into pairs, we can tackle each pair separately and attempt to extract common factors. This process sometimes may require trial and error to rearrange terms effectively, but it often simplifies complex polynomials into more manageable parts.

Here's a quick way to use the grouping method:

• Identify if the polynomial can be grouped by pairing terms with common factors.
• Factor out the greatest common factor from each group.
• Check to see if the remaining expressions share a common factor.
• If they do, factor that out to find the result. If not, the polynomial may not be fully factorable by this method.

Greatest Common Factor

The greatest common factor (GCF) is a key concept in factoring polynomials, representing the largest number or expression that can be divided evenly into all the terms in a polynomial. Finding the GCF is crucial when using the grouping method, as it helps simplify the expression and separate it into factors.

To find the GCF of terms in a polynomial:

• Determine the common factors between the terms. These can include constants (like numbers) and variables.
  • For constant numbers, find the largest number that divides each coefficient.
  • For variables, identify the lowest power of the variable that appears in each term.
• Multiply these common factors together to get the GCF.

In the exercise, terms like 36ak and -8ah share the factor 8a, while -27bk and 6bh have a GCF of -3b.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation (such as addition, subtraction, multiplication, or division). Expressions can vary greatly in complexity and form the foundation of much of algebra as they represent real-world scenarios in mathematical terms.

Understanding algebraic expressions involves:

• Identifying the variables and their coefficients, which are the numerical parts of terms.
• Recognizing how like terms (terms with the same variable and power) can be combined or factored.
• Simplifying expressions by applying rules such as associativity and distributivity.

In the original exercise, the algebraic expression 36ak - 8ah - 27bk + 6bh was the starting point. Breaking it into pairs, it transforms into a more manageable form using the principles of algebra and the grouping method.