Problem 18 - A Graphical Approach to College Algebra

Question

Factor each polynomial by grouping. $$x^{3}+3 x^{2}-5 x-15$$

Step-by-Step Solution

Answer

The polynomial \(x^3 + 3x^2 - 5x - 15\) factors to \((x + 3)(x^2 - 5)\).

Step 1: Group Terms

The given polynomial is \(x^3 + 3x^2 - 5x - 15\). First, group the terms into two pairs: \((x^3 + 3x^2) - (5x + 15)\).

Step 2: Factor Out Common Factors

Factor out the greatest common factor (GCF) from each pair. From the first pair \((x^3 + 3x^2)\), factor out \(x^2\) to get \(x^2(x + 3)\). From the second pair \(-5x - 15\), factor out \(-5\) to get \(-5(x + 3)\).

Step 3: Identify and Factor the Common Binomial

Both terms now have \((x + 3)\) as a common factor: \(x^2(x + 3) - 5(x + 3)\). Factor out the common binomial \((x + 3)\) to get \((x + 3)(x^2 - 5)\).

Step 4: Final Factored Form

The expression is now completely factored and is \((x + 3)(x^2 - 5)\).

Key Concepts

Grouping Method

The grouping method is a handy technique for factoring polynomials, especially when dealing with four terms. It involves rearranging and grouping terms to make the factoring process simpler. In our given polynomial, \(x^3 + 3x^2 - 5x - 15\), the process begins by pairing terms cautiously.To apply the grouping method:

• First, assess the polynomial for pairs of terms that can be grouped together. These pairs should ideally give you a common factor when factored individually.
• With \(x^3 + 3x^2\), and \(-5x - 15\), notice how each group can be separately managed for factorization.

The goal is to manipulate the polynomial into a form that lets you easily spot and extract a common factor from each respective group. This sets the stage for simplifying the polynomial further.

Greatest Common Factor

Finding the greatest common factor (GCF) is crucial in the grouping method. It helps simplify each term in your polynomial and pulls out common factors for easier management.Here's how you find the GCF:

• Look at each pair of terms. For \(x^3 + 3x^2\), the GCF is \(x^2\) since both terms contain \(x^2\).
• In the pair \(-5x - 15\), the GCF is \(-5\). Common factors are valuable because they simplify expressions and assist in spotting further factoring possibilities.

Extracting the GCF from each pair transforms the terms into a more manageable format. Ensuring that both pairs reveal a common binomial sets up the polynomial for the final steps.

Factored Form

Achieving the factored form means breaking down the polynomial into its simplest parts. For our polynomial \(x^3 + 3x^2 - 5x - 15\), applying the grouping method leads to seeing it in a factored form.After extracting the GCF:

• The terms \(x^2(x + 3)\) comes from \(x^3 + 3x^2\), and \(-5(x + 3)\) emerges from \(-5x - 15\).
• Notice both terms include \((x + 3)\), which allows you to further factor the expression.

By factoring \((x + 3)\) from \(x^2(x + 3) - 5(x + 3)\), you simplify the polynomial to \((x + 3)(x^2 - 5)\). This is the fully factored form, showing all component factors succinctly and providing a neat solution. This final form succinctly highlights how complex polynomials can be simplified through careful application of grouping and factoring techniques.