Problem 74 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$40 j^{3}+72 j k-55 j^{2} k-99 k^{2}$$

Step-by-Step Solution

Answer

The factored form of the given expression using the grouping method is: \( 40 j^{3}+72 j k-55 j^{2} k-99 k^{2} = (5j^2+9 k)(8j - 11k) \).

Step 1: Group the terms

In this step, rearrange and group the terms into pairs. Here, we can group the first two terms and the last two terms: $$ (40 j^3+72 j k) + (-55 j^2 k - 99 k^2) $$

Step 2: Factor out the GCD of each group

Next, we need to factor out the GCD from each group. For the first group, the GCD is \(8j\), and for the second group, the GCD is \(-11k\). After factoring out the GCDs, we have: $$8j (5 j^2+9 k) - 11k (5j^2+9 k)$$

Step 3: Factor out the common binomial factor

Now, we notice that both terms have the common binomial factor \((5j^2+9 k)\). Factor out this term: $$ (5j^2+9 k)(8j - 11k) $$ The factored form of the given expression using grouping method is: $$ 40 j^{3}+72 j k-55 j^{2} k-99 k^{2} = (5j^2+9 k)(8j - 11k) $$

Key Concepts

Grouping Method

The grouping method is a useful strategy for factoring polynomials, especially when dealing with four terms. This method involves rearranging and grouping the terms into smaller sets (usually pairs), so that each set can be factored separately.
The aim is to find a common factor in each group, which can then be used to simplify the entire expression. To apply the grouping method:

• Identify terms that can be paired. Look for terms with something common.
• Group these terms together using parentheses, making sure to consider signs.
• Factor out the greatest common divisor (GCD) from each group to simplify it.
• Finally, look for a common factor across the new groups and factor it out.

This method is quite helpful in making complex polynomials easier to work with by breaking them into manageable parts.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying expressions, particularly in algebra.
The GCD of a set of numbers or terms is the largest factor that divides all the numbers or terms without leaving a remainder. In the context of factoring polynomials by the grouping method, it aids in simplifying each group effectively. To find the GCD:

• List all the factors of each term or number involved.
• Identify the largest factor that appears in all terms.
• Factor this out from the group, simplifying the expression.

In the provided exercise, factoring the GCD from each group was crucial in simplifying and eventually factoring the entire polynomial expression. For the first group, the GCD was \(8j\), extracted from each term in that group, while the second group had a GCD of \(-11k\). Factoring these out made it easier to spot the common binomial factor in the succeeding steps.

Binomial Factor

A binomial factor plays a crucial role when factoring expressions like polynomials, as seen in the final steps of our exercise.
A binomial is simply an algebraic expression that contains two terms, which are combined together by addition or subtraction.
Sometimes in polynomials, the same binomial may appear in multiple terms, allowing it to be factored out.Factoring binomial expressions consists of the following:

• Identify the binomial that repeats or is common between terms.
• Factor this binomial out to simplify the expression.

In the exercise, the binomial \((5j^2 + 9k)\) was common between the factored groups.
By identifying and factoring it out, we simplified the polynomial to its final factored form: \((5j^2 + 9k)(8j - 11k)\). Finding the binomial common factor often helps to streamline polynomials and solve equations more easily.

Algebraic Expressions

Algebraic expressions are combinations of numbers, letters (variables), and mathematical operations.
They form the basis of algebra and can range from simple combinations like \(x + 1\) to more complex polynomials like the one in our exercise.Key components of algebraic expressions include:

• Terms: Each separate part of an expression, typically separated by `+` or `-` signs.
• Coefficients: Numbers that are multiplied by variables within terms.
• Variables: Letters that represent numbers and can change in value.
• Operators: Symbols like `+`, `-`, `×`, or `÷` that indicate operations.

Understanding how to manipulate algebraic expressions by applying techniques such as factoring helps solve problems not just in algebra, but in various fields like calculus and trigonometry as well.
In our exercise, these concepts helped in organizing the polynomial so it could be effectively factored, using methods like grouping and recognizing common factors.