Problem 17 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 24 k m p+6 k p^{2}+40 m p+10 p^{2} $$

Step-by-Step Solution

Answer

(4m + p)[2p(3k + 5)]

Step 1 - Group the Terms

Group the terms in pairs to make factoring easier: \( (24 k m p + 6 k p^2) + (40 m p + 10 p^2) \)

Step 2 - Factor Out the Greatest Common Factor (GCF) from Each Pair

Factor out the GCF from each group: \( 6kp(4m + p) + 10p(4m + p) \)

Step 3 - Factor Out the Common Binomial

Both terms contain the common binomial \((4m + p)\). Factor out \((4m + p)\): \((4m + p)(6kp + 10p)\)

Step 4 - Simplify the Expression

Finally, simplify the second part of the expression: \((4m + p)[2p(3k + 5)]\)

Key Concepts

Grouping Terms

Grouping terms is the initial step in factoring polynomials, especially when dealing with four or more terms. By organizing the polynomial into smaller groups, we can more easily identify common factors within each group. In our given polynomial, we start by grouping the terms in pairs:
\((24 k m p + 6 k p^2) + (40 m p + 10 p^2)\).
This helps us to focus on smaller sections of the polynomial, making it simpler to factor. Think of it like breaking down a complex problem into manageable chunks.

Factoring Out the Greatest Common Factor (GCF)

After grouping the terms, the next step is to factor out the Greatest Common Factor (GCF) from each grouped pair. The GCF is the highest number and variable(s) that can evenly divide each term in the group.
For the first group, \(24 k m p + 6 k p^2\), the GCF is \(6 k p\), so we factor it out:
\ 6kp(4m + p)\
For the second group, \(40 m p + 10 p^2\), the GCF is \(10p\), so we factor it out:
\ 10p(4m + p)\
This step simplifies each group, making it easier to identify further common factors.

Factoring Out the Common Binomial

Once we have factored out the GCF from each grouped pair, we often find a common binomial. In our example, both groups contain the binomial \((4m + p)\). This common factor can be factored out again to simplify the polynomial even more. Here’s how it looks:
\ (4m + p)(6kp + 10p)\
Recognizing a common binomial factor is crucial. It allows us to combine the groups into a product of binomials, greatly simplifying the expression.

Simplifying Expressions

The final step in our factoring process is to simplify the expression as much as possible. In the example, we focus on simplifying \(6kp + 10p\) within the binomial product. We notice that \(2p\) is a common factor in \(6kp + 10p\), so we factor it out:
\( (4m + p) [2p(3k + 5)] \).
Simplifying expressions in this way not only makes them easier to work with, but also more understandable, revealing the underlying structure of the polynomial. By following these steps, we ensure that our expression is as simple and clear as possible.