Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ 2 x(x+y)^{2}-8 x\left(x+y^{2}\right)^{2} $$

Step-by-Step Solution

Answer

The factored form of the given polynomial is: \( x\left[ 2 (x+y)^{2} - 8 \left(x+y^{2}\right)^{2} \right] \).

Step 1: Identify common factors

In the given polynomial: $$ 2 x(x+y)^{2}-8 x\left(x+y^{2}\right)^{2} $$ We can see that both terms have a common factor of x. Let's factor out x from both terms.

Step 2: Factor out the common factor

Factoring out x from both terms, we get: $$ x\left[ 2 (x+y)^{2} - 8 \left(x+y^{2}\right)^{2} \right] $$

Step 3: Check if further factoring is possible

Now, we have: $$ x\left[ 2 (x+y)^{2} - 8 \left(x+y^{2}\right)^{2} \right] $$ We can identify no further common factors within the brackets, and the expressions are not easily factorable. Thus, we can conclude that the factored form of the given polynomial is: $$ x\left[ 2 (x+y)^{2} - 8 \left(x+y^{2}\right)^{2} \right] $$

Key Concepts

Common Factors in Polynomials

To understand polynomials, one must begin by identifying common factors. Common factors are terms that appear in each part of the polynomial. Much like finding the greatest common denominator in a fraction, finding common factors in polynomials simplifies them.

Identifying Common Factors

When examining a polynomial, we look for terms that are shared across all parts of the expression. For example, in the polynomial given in the exercise, \( 2x(x+y)^2 - 8x(x+y^2)^2 \), the letter \( x \) is a common factor.

Extracting this factor simplifies the polynomial into a more manageable form and is often the first step in the factoring process. It's essential to inspect each term carefully to ensure no common factors are overlooked, as missing them can make the rest of the factoring process more complex, or even lead to false assumptions about the polynomial being prime.

Polynomial Factorization Techniques

Factorization of polynomials is a vital algebraic skill that can help to simplify expressions and solve equations. Polynomials can be factored using a variety of techniques, depending on their structure and complexity.

Different Techniques for Factoring

Once common factors are factored out, other techniques such as the difference of squares, sum and difference of cubes, or factoring by grouping may be applied. One must also look out for patterns that resemble special products.

In the given exercise, after factoring out \( x \), what remains does not present itself to conventional methods, hence indicating that it may be a prime polynomial or require more advanced techniques like completing the square or quadratic formula in case of quadratic polynomials to simplify further. A good tip is to always simplify what can be simplified first, using the easiest methods before moving on to more complex ones.

Prime Polynomials Identification

Pondering over prime polynomials is like searching for a hidden door that doesn't exist. Just as prime numbers have no divisors other than 1 and themselves, prime polynomials cannot be factored into the product of two non-constant polynomials.

Recognizing Indivisibility

A prime polynomial is one that stands indivisible after all factoring techniques have been exhausted. In our example \( x[2(x+y)^2 - 8(x+y^2)^2] \), after removing the common factor of \( x \), we reach a stage where no further obvious factoring is possible.

Identifying a prime polynomial requires one to challenge all possible factorization routes and reach a conclusive impossibility. This process may be intuitive for simple expressions but can become quite intricate with higher-degree polynomials. Additionally, knowing that factoring over the integers may fail while it could be possible over other fields (such as complex numbers) is crucial for a comprehensive understanding of the concept of prime polynomials.