Problem 27 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The polynomial is factored as \((2x - 3)(x^{2} + 1)\).

Step 1: Group the terms

Take the polynomial \(2x^{3} - 3x^{2} + 2x - 3\) and group the first two terms and the last two terms. This gives us: \((2x^{3} - 3x^{2}) + (2x - 3)\).

Step 2: Factor out the greatest common factor (GCF) of each group

For the first group \((2x^{3} - 3x^{2})\), the GCF is \(x^{2}\). Factoring out \(x^{2}\), we get \(x^{2}(2x - 3)\). For the second group \((2x - 3)\), the GCF is 1, so it remains \(1(2x - 3)\).

Step 3: Factor by grouping

Notice that \((2x - 3)\) is a common factor in both groups. Factor \((2x - 3)\) out of the expression. This gives us \((2x - 3)(x^{2} + 1)\).

Step 4: Conclusion: Verify the factorization

Verify the factorization by expanding \((2x - 3)(x^{2} + 1)\). If it gives back the original polynomial, the factorization is correct. Expanding gives \(2x^{3} + 2x - 3x^{2} - 3\), which indeed equals the original polynomial.

Key Concepts

Grouping Method

Factoring polynomials can be challenging, but breaking them into manageable chunks can simplify the process. The grouping method is a great strategy to achieve this. Essentially, it involves grouping terms with common factors within a polynomial expression so they can be factored separately.

When dealing with a polynomial like \(2x^{3} - 3x^{2} + 2x - 3\), you'll group terms strategically. In this example, terms \((2x^{3} - 3x^{2})\) and \((2x - 3)\) are grouped. This allows us to see the expression in simpler terms, enabling the next stage of factorization.

• Identify pairs of terms that share a common factor.
• Group these pairs using parentheses.
• Make sure the grouped pairs can easily be factored further.

Applying the grouping method is akin to solving a jigsaw puzzle, where the right grouping leads to the key to factorization.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that two or more terms have in common. It's a pivotal concept when factoring polynomials using the grouping method.

After grouping, identify the GCF in each group. In the example polynomial, \(2x^{3} - 3x^{2}\) has a GCF of \(x^{2}\), and for \(2x - 3\), the GCF is simply 1. Knowing how to find the GCF is essential as it simplifies each group, bringing you closer to a factored expression.

• For multiple variables, include the lowest power of each common variable.
• Factor out the GCF to simplify each group.
• Check your groups to ensure the remaining expression is simple and manageable.

This approach not only simplifies the expression but brings us to the final step of finding the common factor across groups.

Polynomial Expressions

A polynomial expression is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient. Understanding their structure aids in efficiently factoring them.

The process of factoring involves breaking down the polynomial into simpler terms or products of terms that make it up, such as \((2x - 3)(x^{2} + 1)\) from the given polynomial.

• Identify the degree of the polynomial – the highest power of the variable present.
• Understand that every polynomial term can be added, subtracted, or rearranged.
• Recognize patterns and factors common within the terms for simplification.

Grasping the essence of polynomial expressions allows for a more intuitive approach to factorization and problem-solving, making algebra more manageable.