Problem 84 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 6 n^{2} p+3 n^{2} w-54 p-27 w $$

Step-by-Step Solution

Answer

\( (2p + w)3(n + 3)(n - 3) \)

Step 1 - Group the terms

Group the terms into two pairs that can be factored separately. \[ 6n^2 p + 3n^2 w - 54p - 27w = (6n^2 p + 3n^2 w) + (-54p - 27w) \]

Step 2 - Factor out the greatest common factor (GCF) from each group

Factor out the GCF from each group. For the first group, \( 6n^2 p + 3n^2 w \): \[ 3n^2 (2p + w) \]For the second group, \( -54p - 27w \): \[ -27 (2p + w) \]Rewrite the expression now: \[ 3n^2 (2p + w) - 27 (2p + w) \]

Step 3 - Factor out the common binomial factor

The expression \( 3n^2 (2p + w) - 27 (2p + w) \) has a common binomial factor \( 2p + w \). Factor out this binomial: \[ (2p + w)(3n^2 - 27) \]

Step 4 - Factor the remaining expression if possible

Check if the remaining expression can be factored further. The term inside the parenthesis \( 3n^2 - 27 \) has a common factor: \[ 3(n^2 - 9) \]Notice that \( n^2 - 9 \) is a difference of squares: \[ n^2 - 9 = (n + 3)(n - 3) \]So the factorization is: \[ 3(n + 3)(n - 3) \]Therefore, the complete factorization of the original expression is: \[ (2p + w)3(n + 3)(n - 3) \]

Key Concepts

Greatest Common Factor

When factoring polynomials, identifying the Greatest Common Factor (GCF) is crucial. The GCF is the largest factor that divides two or more terms. For example, consider the terms in the original problem:

• 6n²p and 3n²w
• -54p and -27w

We start by factoring each group. For 6n²p and 3n²w, the GCF is 3n². For -54p and -27w, the GCF is -27. Factoring those out, you get:

• 6n²p + 3n²w = 3n²(2p + w)
• -54p - 27w = -27(2p + w)

Grouping Terms

Grouping terms is a method used to simplify polynomials by combining like terms. In this problem, group terms into pairs that have a common factor. For example:

• Group 1: 6n²p + 3n²w
• Group 2: -54p - 27w

This helps to more easily identify the GCF in smaller segments, simplifying our next steps. After factoring out the GCF, the equation simplifies to:

• 3n²(2p + w) - 27(2p + w)

Difference of Squares

The Difference of Squares is a specific polynomial form:

• [ a² - b² = (a + b)(a - b) ]

In this problem, 3(n² - 9) can be converted using the Difference of Squares. Notice:

• [ n² - 9 = (n + 3)(n - 3) ]

So the problem becomes 3(n + 3)(n - 3). This step ensures the polynomial is completely factored.

Binomial Factors

A binomial factor is a polynomial with exactly two terms. In this problem, the common binomial factor is (2p + w). Factoring this out helps us simplify the expression further. After grouping terms and factoring out their GCFs, we write the expression:

• [ 3n²(2p + w) - 27(2p + w) = (2p + w)(3n² - 27) ]

Finally, incorporating the Difference of Squares, we get the complete factorization: (2p + w)3(n + 3)(n - 3).