Problem 30 - College Algebra with Modeling and Visualization

Question

Use grouping to factor the polynomial. \(2 x^{4}-5 x^{3}+10 x-25\)

Step-by-Step Solution

Answer

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

Step 1: Group the Terms

We begin by grouping the terms in pairs to explore the possibility of factoring by grouping. Group the polynomial into two parts: \((2x^4 - 5x^3) + (10x - 25)\).

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

Look for the greatest common factor in each group. For the first group, \(2x^4 - 5x^3\), the GCF is \(x^3\), and for the second group, \(10x - 25\), the GCF is \(5\). Factor each group:- From \(2x^4 - 5x^3\), factor out \(x^3\) to get \(x^3(2x - 5)\).- From \(10x - 25\), factor out \(5\) to get \(5(2x - 5)\).

Step 3: Identify the Common Binomial

After factoring both groups, we have the expression:\(x^3(2x - 5) + 5(2x - 5)\).Notice that \((2x - 5)\) is a common binomial factor in both terms.

Step 4: Factor Out the Common Binomial

Since \((2x - 5)\) is common to both terms, factor it out:\((x^3 + 5)(2x - 5)\).

Step 5: Verify the Factorization

Double-check the factorization by expanding \((x^3 + 5)(2x - 5)\) to ensure that it results in the original polynomial. Expand using the distributive property:- First, \(x^3 \times 2x = 2x^4\).- Then, \(x^3 \times (-5) = -5x^3\).- Next, \(5 \times 2x = 10x\).- Finally, \(5 \times (-5) = -25\).Thus, the original polynomial \(2x^4 - 5x^3 + 10x - 25\) is indeed equivalent to \((x^3 + 5)(2x - 5)\).

Key Concepts

Grouping Method

When you encounter a polynomial that seems complex, the grouping method offers a structured way to simplify it. It involves organizing terms into pairs or groups that can be factored. Start by looking at a polynomial's terms and splitting them into two parts. Here is an example: for the polynomial \(2x^4 - 5x^3 + 10x - 25\), divide it into

• \((2x^4 - 5x^3)\): The first group
• \((10x - 25)\): The second group

How it Works:

• Look for common factors within each group.
• Re-factor to uncover any repeated patterns or terms.
• Often, this technique reveals a common binomial factor, simplifying the expression.

The grouping method is especially useful when traditional factoring doesn't immediately work. It provides another angle to break down complex expressions into manageable parts.

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial concept for simplifying expressions. It's the largest factor that divides two or more terms without leaving a remainder. By identifying and factoring out the GCF, you reduce the polynomial into simpler, smaller forms.Steps to Find the GCF:

• Examine each term in the group separately.
• For \(2x^4 - 5x^3\), the GCF is \(x^3\).
• For \(10x - 25\), the GCF is \(5\).

Applying the GCF efficiently simplifies each group in the polynomial significantly. In our example, factoring these GCFs leads to \(x^3(2x - 5)\) and \(5(2x - 5)\). By removing these common factors, we encourage simplification further downstream in the factoring process.

Binomial Factor

A binomial factor is a two-term expression that appears throughout a polynomial. Recognizing and factoring out a binomial can drastically simplify the polynomial's form. Once the terms are grouped and the GCF is extracted, look for similar terms within those results.Example:

• In \((x^3(2x - 5) + 5(2x - 5))\), \((2x-5)\) is a common binomial factor.

By identifying \((2x-5)\) as common here, you can factor further. Express the polynomial now as \((x^3 + 5)(2x - 5)\). This step consolidates the expression by leveraging repeated patterns, simplifying the polynomial dramatically. It's a key step in the factoring journey, leading to the simplified binomial form.

Distributive Property

The distributive property is a fundamental aspect of algebra, enabling the multiplication of an entire expression by a single term. This property aids in expanding or simplifying expressions. Verification:

• Use it to check if factoring was done correctly.
• Multiply each term: \(x^3 \times 2x = 2x^4\) and \(x^3 \times (-5) = -5x^3\).
• Similarly, \(5 \times 2x = 10x\) and \(5 \times (-5) = -25\).

Combining these terms demonstrates the polynomial's original form: \(2x^4 - 5x^3 + 10x - 25\). This verification signifies that the factorization is accurate. Using the distributive property ensures reliability in your factoring results and reinforces your knowledge of algebraic operations.