Problem 77 - Elementary Algebra

Question

Use any of the factoring methods to factor. Identify any prime polynomials. $$ 9 m^{2}+81 n^{2} $$

Step-by-Step Solution

Answer

The polynomial factors to \( 9(m^2 + 9n^2) \) and \( m^2 + 9n^2 \) is prime.

Step 1: Identify the Form

Recognize that the given polynomial is of the form \[ 9m^2 + 81n^2 \]. This is not a simple difference of squares since it is a sum.

Step 2: Find Common Factors

Determine if there is a common factor that can be factored out from both terms. Notice that both terms have the common factor of 9. Thus, rewrite the polynomial as:\[ 9(m^2 + 9n^2) \].

Step 3: Check for Further Factoring

Examine the expression inside the parentheses: \(m^2 + 9n^2\). This is a sum of squares and there is no straightforward factoring for a sum of squares using real numbers.

Step 4: Determine if the Polynomial is Prime

Since the sum of squares \(m^2 + 9n^2\) cannot be factored further over the real numbers, it is considered a prime polynomial.

Key Concepts

Common Factors

When factoring polynomials, the first step is often to identify any common factors among the terms. Common factors are factors that are shared by all the terms in a polynomial. For the given expression \(9m^2 + 81n^2\), both terms share a common factor of 9.
This initial step simplifies the polynomial by factoring out the greatest common factor (GCF).

• Rewrite the polynomial as: \[ 9(m^2 + 9n^2) \].

The GCF of 9 is

Sum of Squares

Another core concept in factoring is recognizing different forms of polynomials, such as sums of squares. Unlike the difference of squares \((a^2 - b^2)\), a sum of squares \((a^2 + b^2)\) doesn’t factor into real numbers. For our expression inside the parenthesis \(m^2 + 9n^2\), we see it is a sum of squares. This sum of squares cannot be simplified further using real numbers, although complex factoring is possible.
The important takeaway is:

• Sums of squares do not factor over the real numbers.

Prime Polynomials

A polynomial is considered prime if it cannot be factored further over the real number system. In our example, the expression \(m^2 + 9n^2\) is a prime polynomial. This is because a sum of squares like this cannot be factored using real numbers.
Prime polynomials are important since they represent the simplest form of a polynomial, meaning they cannot be broken down into simpler polynomial forms.
Here’s what it means for a polynomial to be prime:

• It has no further factorization in the set of real numbers.

This highlights the importance of recognizing when a polynomial has reached its simplest form.