Problem 66 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 8 w^{2}+7 w v+8 w+7 v $$

Step-by-Step Solution

Answer

The polynomial factors as \( (8w + 7v)(w + 1) \).

Step 1: Group the Terms

Separate the polynomial into two binomials. Group the first two terms together and the last two terms together. They should look like this: \( (8w^2 + 7wv) + (8w + 7v) \).

Step 2: Factor Each Group Separately

Factor out the greatest common factor from each group. For the first group \( (8w^2 + 7wv) \), factor \( w \): \[ w(8w + 7v) \].For the second group \( (8w + 7v) \), there is no factor to factor out.

Step 3: Factor Out the Common Binomial

Both groups contain the common binomial \( (8w + 7v) \). Factor out \( (8w + 7v) \) from the entire expression: \[ (8w + 7v)(w + 1) \].

Key Concepts

Grouping Method

The grouping method is a handy technique used when you're tasked with factoring polynomials, especially those with four or more terms. It works by organizing the terms of a polynomial into smaller manageable groups, which can then be factored individually. Here’s how you can understand the process:First, you take the original polynomial and split it into two separate groups. Generally, you pair the first two terms together and the last two terms together. This is often the most logical breakdown. For example, let's look at the polynomial: \( 8w^2 + 7wv + 8w + 7v \).By using the grouping method, you split it as:* \((8w^2 + 7wv) + (8w + 7v)\)Each of these small groups is much easier to manage, allowing you to further factorize them down the line. This technique simplifies the polynomial, making it much more approachable to find common factors and factored forms.

Common Factor

A common factor is a term that appears in multiple polynomial terms, and it's crucial for simplifying polynomial expressions. When you have grouped your terms using the grouping method as earlier mentioned, you can more easily spot any potential common factors in each group.Consider the groups from our example:* \(8w^2 + 7wv\)* \(8w + 7v\)In the first group, \(8w^2 + 7wv\), the common factor is \(w\), because both terms in this group contain \(w\). Therefore, we can factor \(w\) out:* \(w(8w + 7v)\)Meanwhile, the second group, \(8w + 7v\), does not have any common factors, which means it remains as it is.Finding common factors in groups helps to simplify complex polynomials, reducing them to a product of simpler polynomials, facilitating easier further operations or solutions.

Binomials

Binomials are algebraic expressions containing exactly two unlike terms, separated by a plus or minus sign. Recognizing binomials is vital in the process of factoring polynomials by grouping, particularly when you're looking to factor out a common binomial factor.Let's refer back to our example polynomial:* \((8w^2 + 7wv) + (8w + 7v)\)After factoring out the common term from each group, we noticed:* \(w(8w + 7v) + (8w + 7v)\)Here, \((8w + 7v)\) is the common binomial between the two products. Identifying such common binomials is essential because it allows you to factor the entire polynomial further. The expression can now be rewritten as a product of binomials:* \((8w + 7v)(w + 1)\)Understanding binomials and being able to factor them out means simplifying and solving polynomials becomes significantly more manageable, bringing complex expressions into a simpler realm.