Problem 83 - College Algebra

Question

In Exercises \(57-84\), factor completely, or state that the polynomial is prime. $$2 x^{3}-8 a^{2} x+24 x^{2}+72 x$$

Step-by-Step Solution

Answer

The completed factorization of the polynomial is: \(2x\left(x^{2} + 12x - 4[(a - 3)(a + 3)]\right)\.

Step 1: Isolate similar terms

Group the terms which have common factors. This can be done as \(2x^{3} + 24x^{2}\) and \(-8a^{2}x + 72x\).

Step 2: Factor out the Common from each Group

Take out the greatest common factor from each group: \(2x^{2}(x + 12) - 8x(a^{2} - 9)\.

Step 3: Factor out the Common Factor

Identify the common factor in the both terms and factor it out to obtain the final result: \(2x[(x + 12)x - 4(a^{2} - 9)]\.

Step 4: Factor Completely

Continue to factor the quadratic expression \(a^{2} - 9\) as \((a - 3)(a + 3)\) and hence the final result: \(2x\left(x^{2} + 12x - 4[(a - 3)(a + 3)]\right)\.

Key Concepts

Greatest Common Factor

In polynomial factorization, the first step often involves identifying the greatest common factor (GCF). The GCF is the largest expression that divides all of the terms in a polynomial with no remainder. For example, consider the polynomial \(2x^{3} + 24x^{2} + -8a^{2}x + 72x\). Here, each term contains the variable \(x\).

• The GCF of \(2x^{3}\) and \(24x^{2}\) is \(2x^{2}\).
• The GCF of \(-8a^{2}x\) and \(72x\) is \(-8x\).

Extracting these common factors is crucial because it simplifies the polynomial and paves the way for further factorization. Always start by finding and factoring out the GCF to make subsequent steps easier.

Grouping Method

The grouping method is a technique for factoring polynomials, especially useful when dealing with four or more terms. This method entails rearranging the polynomial and grouping terms to find common factors. Let's examine it applied to our polynomial:

• First, group terms with similar factors: \( (2x^{3} + 24x^{2}) + (-8a^{2}x + 72x) \).
• Next, factor out the GCF from each group: from \(2x^{2}\) and \(x + 12\); from \(-8x\) and \(a^{2} - 9\).

With the expressions factored in pairs, it becomes easier to identify a common binomial factor, which simplifies the polynomial into its factors. This method effectively reduces the complexity of polynomial expressions.

Factoring Quadratics

Factoring quadratics is a foundational skill in algebra, applicable when polynomials are of the form \(ax^{2} + bx + c\). Although the original polynomial provided isn't a direct quadratic, it contains quadratic-like terms within its sub-expressions.
One notable pattern is the difference of squares, which we can apply to expressions like \(a^{2} - 9\). It's factored as:

• Recognize it as \(a^{2} - 3^{2}\) to factor as \((a - 3)(a + 3)\).

Identifying and applying patterns, such as factoring quadratics or differences of squares, helps break down complex polynomials into simpler, manageable components.

Prime Polynomials

In algebra, a prime polynomial is a polynomial that cannot be factored into the products of polynomials with integer coefficients. Identifying a prime polynomial involves checking if further factorization is possible.
In our exercise, after all the steps of grouping and factoring, we continuously monitor whether we can factor it further.
The notion of a prime polynomial reminds us that not all polynomials are reducible using simple factorization. If a polynomial remains with no common factors and is not factored into simpler binomials or monomials, it's considered prime. Understanding when a polynomial cannot be simplified further is as vital as the ability to factor it.