Problem 29 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The factored form is \((2x - 1)(x^3 + 2)\).

Step 1: Group Terms

First, observe the given polynomial: \(2x^4 - x^3 + 4x - 2\). We will group these terms in pairs to facilitate factoring by grouping.Group them as: \((2x^4 - x^3) + (4x - 2)\).

Step 2: Factor Out the Greatest Common Factor in Each Group

In the first group \((2x^4 - x^3)\), the greatest common factor is \(x^3\). Factor it out to get \(x^3(2x - 1)\).In the second group \((4x - 2)\), the greatest common factor is \(2\). Factor it out to get \(2(2x - 1)\).

Step 3: Factor the Entire Expression

After factoring each group, the expression is: \(x^3(2x - 1) + 2(2x - 1)\).Notice that \((2x - 1)\) is a common factor in both terms.Factor \((2x - 1)\) out of the entire expression: \((2x - 1)(x^3 + 2)\).

Step 4: Write the Final Factored Form

The polynomial \(2x^4 - x^3 + 4x - 2\) has been factored by grouping as \((2x - 1)(x^3 + 2)\). This is its simplest factored form.

Key Concepts

Using Grouping Method to Factor Polynomials

Factoring polynomials can be challenging at times, especially with complex expressions. The grouping method makes it easier by breaking the problem into simpler parts. In this approach, we group terms of the polynomial to take advantage of patterns that simplify the factoring process.
For example, consider a polynomial with four terms like the one in our original exercise: \(2x^4 - x^3 + 4x - 2\).

• First, we group terms in pairs, aligning similar terms. This might involve rearranging the equation if necessary.
• In our case, the polynomial can be grouped as \((2x^4 - x^3) + (4x - 2)\).
• This setup helps us focus on smaller parts of the equation, making the factoring process more manageable.

By grouping, we turn a complex problem into a series of simpler ones that can be addressed step by step.

Finding the Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) plays a pivotal role in factoring polynomials, as it helps in simplifying terms within each group. The GCF is the largest factor that divides all terms in a group without leaving a remainder.
Finding the GCF in each group involves:

• Identifying the highest power of common variables shared by each term.
• Factoring out these variables along with any numerical coefficients shared by the terms.

Let's revisit our grouped terms \((2x^4 - x^3) + (4x - 2)\):

• In the first group, the GCF is \(x^3\). Factoring it out, we get \(x^3(2x - 1)\).
• In the second group, the GCF is \(2\). Factoring it out, we get \(2(2x - 1)\).

Identifying and factoring out the GCF simplifies each group, setting the stage for the next phase of the process.

Achieving Factored Form

The ultimate aim when factoring a polynomial is to achieve its factored form. This simplifies further algebraic operations and solves equations efficiently. After factoring out the GCF from each group, we are often left with a common factor that appears across all the groups. This commonality is key to taking the final step in factoring by grouping. In our polynomial, \(x^3(2x - 1) + 2(2x - 1)\), notice that \((2x - 1)\) appears in both terms:

• Factor this common term \((2x - 1)\) out of the entire expression. This gives the expression its final factored form.

Thus, the polynomial \(2x^4 - x^3 + 4x - 2\) factors neatly into \((2x - 1)(x^3 + 2)\). This expression is much simpler to work with, whether you're looking to solve an equation or analyze its properties.