Problem 53 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 3 n^{2} x-12 n^{2}+m n x-4 m n $$

Step-by-Step Solution

Answer

The factored form is \((x - 4)(3n^2 + mn)\).\(3n^2 + mn\) is a prime polynomial.

Step 1: Group the terms in pairs

Group the given polynomial terms in pairs to simplify the factorization process.yields: \( (3n^2x - 12n^2) + (m n x - 4 m n) \)

Step 2: Factor out the greatest common factor (GCF) from each pair

Factor out the GCF in each grouped term. For the first group \(3n^2(x - 4)\) and for the second group \(mn(x - 4)\).So now we have:\( 3n^2(x - 4) + mn(x - 4) \)

Step 3: Factor out the common binomial

Notice that \((x - 4)\) is a common factor in both terms. Factor \((x - 4)\) out.Resulting in:\( (x - 4)(3n^2 + mn) \)

Step 4: Check for any further factorization or prime polynomial

Assess whether \((3n^2 + mn)\) can be factored further. Since no further factorization is possible, it is a prime polynomial.Hence,\( (x - 4)(3n^2 + mn) \) is the complete factorization.

Key Concepts

greatest common factor

When factoring a polynomial, a great starting point is to find the greatest common factor (GCF) for each term group. The GCF is the largest factor that each term in a pair shares. In our example, the polynomial is grouped into two pairs: \((3n^2x - 12n^2)\) and \((mnx - 4mn)\). For the first pair, the GCF is \(3n^2\), so factor it out:

• \(3n^2(x - 4)\)
  

For the second pair, the GCF is \(mn\), thus:

• \(mn(x - 4)\)
  

Finding the GCF simplifies polynomials by grouping similar factors, which makes the next steps in factorization more straightforward.

binomial factorization

After determining the GCF, the next key step is to factor out any common binomials. This step makes it easier to combine terms and build a simpler expression. In our example, after extracting the GCF from each pair, we have:

• \(3n^2(x - 4) + mn(x - 4)\)
  

Notice that \((x - 4)\) is a common binomial factor in both terms. Therefore, factor \((x - 4)\) out:

• \((x - 4)(3n^2 + mn)\)
  

Identifying and factoring out a common binomial can greatly simplify the polynomial.

prime polynomial

The final step in polynomial factorization involves checking if the resulting expression can be simplified any further. If not, the resulting polynomial is considered a prime polynomial. A prime polynomial cannot be factored into simpler polynomials with integer coefficients. In our example, after factoring out the binomial \((x - 4)\), we have:

• \((x - 4)(3n^2 + mn)\)
  

Upon inspecting \(3n^2 + mn\), there are no further common factors or methods for further factorization with integer coefficients. Hence, \(3n^2 + mn\) is a prime polynomial, and the complete factorization of the original expression is:

• \((x - 4)(3n^2 + mn)\)
  

Recognizing prime polynomials ensures that you have fully simplified the expression.

pair grouping

Pair grouping is a useful strategy to break down polynomials for easier factorization. This involves grouping terms that share common factors, making it simpler to factor out the GCF at the start. For the polynomial \(3n^2x - 12n^2 + mnx - 4mn\), we arrange it into pairs:

• \((3n^2x - 12n^2) + (mnx - 4mn)\)
  

This grouping allows us to easily find and extract the GCF for each pair. By dealing with smaller, more manageable pieces, we simplify the process of revealing common factors and binomials. This technique is particularly handy for complex polynomials.