Problem 88 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 5 x^{3}+10 x^{2}-20 x-40 $$

Step-by-Step Solution

Answer

The polynomial factored is \( 5(x + 2)^2(x - 2) \).

Step 1: Identify Common Factors

Look at each term of the polynomial: \[ 5x^3, \ 10x^2, \ -20x, \ -40 \] Notice that 5 is a common factor of all terms. We can factor out 5 from the polynomial. This simplifies our polynomial for the next steps.

Step 2: Factor Out the Greatest Common Factor (GCF)

Factor out the common factor of 5 from the polynomial: \[ 5(x^3 + 2x^2 - 4x - 8) \] Now, we simplify the polynomial to a simpler form that we can factor further.

Step 3: Group Terms for Factoring by Grouping

Group the terms in pairs to find a common binomial factor: \[ 5[(x^3 + 2x^2) + (-4x - 8)] \] This sets us up to factor each pair separately, looking for any further common factors.

Step 4: Factor Each Pair

Factor each pair of terms separately: 1. From \( x^3 + 2x^2 \), factor out \( x^2 \): - \( x^2(x + 2) \)2. From \( -4x - 8 \), factor out \( -4 \): - \( -4(x + 2) \)Both terms now share the common factor \( (x + 2) \).

Step 5: Combine Using Common Factor

Use the common binomial factor \( (x + 2) \) to rewrite as a product of factors:\[ 5[(x^2 - 4)(x + 2)] \] This can now potentially be simplified further.

Step 6: Further Factor the Difference of Squares

Notice that \( x^2 - 4 \) is a difference of squares, which can be factored as \[ (x + 2)(x - 2) \]Thus, the polynomial is now: \[ 5(x + 2)(x + 2)(x - 2) \]

Step 7: Simplify the Expression

Since \( (x + 2) \) appears twice, simplify to show clearly:\[ 5(x + 2)^2(x - 2) \]This is the factored form of the original polynomial.

Key Concepts

Greatest Common Factor

Before diving into complex methods of factorization, it's crucial to start with the Greatest Common Factor (GCF). This simplifies the polynomial by factoring out the most significant term common in all expressions.

**What is the Greatest Common Factor?**
The GCF of a set of terms is the largest factor that divides each of the terms without leaving a remainder. In simpler terms, it's the biggest number and set of variables you can "pull out" from each term.

In our polynomial,

• we have 4 terms: \(5x^3\), \(10x^2\), \(-20x\), and \(-40\).
• The number 5 is the largest number that can divide all coefficients (5, 10, 20, and 40).

By factoring out the GCF, 5, the polynomial is reduced to: \[ 5(x^3 + 2x^2 - 4x - 8) \]

Removing the GCF simplifies further steps in solving the polynomial. It makes spotting other patterns like grouping or special products easier.

Factoring by Grouping

Factoring by grouping involves rearranging and grouping terms to simplify the polynomial further. This method is especially handy for polynomials with four or more terms.

**How Does Factoring by Grouping Work?**
Once the GCF is factored out, break down the polynomial into two pairs: \((x^3 + 2x^2)\) and \((-4x - 8)\).

Here's what happens:

• In the first pair \(x^3 + 2x^2\), factor out \(x^2\). This gives \(x^2(x + 2)\).
• In the second pair \(-4x - 8\), factor out \(-4\). This results in \(-4(x + 2)\).

You'll notice that both pairs share a common factor: \((x + 2)\).

With both groups sharing \((x + 2)\) as a factor, they can now be expressed as: \[ 5[(x^2 - 4)(x + 2)] \]

This method effortlessly simplifies expressions by exploiting internal structures and allows us to solve complex equations more efficiently.

Difference of Squares

The difference of squares is a specific type of polynomial that follows the pattern \(a^2 - b^2 = (a + b)(a - b)\). Identifying this pattern simplifies the factorization significantly.

**Recognizing a Difference of Squares**
You can spot this pattern when two perfect square terms are separated by a subtraction sign. The classic example is \(x^2 - 4\).

• Here, \(x^2\) and \(4\) are both perfect squares, with square roots \(x\) and \(2\) respectively.

**Factoring the Difference of Squares**
Apply the difference of squares formula: \(x^2 - 4 = (x + 2)(x - 2)\).

In our polynomial, once after grouping, we're left with: \(x^2 - 4\). By utilizing the difference of squares, it becomes \((x + 2)(x - 2)\), creating a further breakdown.

The final factorized form exemplifies how recognizing special patterns like the difference of squares aids in reaching the simplest polynomial representation efficiently.