Problem 70 - Algebra A Combined Function

Question

Factor each four-term polynomial by grouping. If this is not possible, write "not factorable by grouping." $$ 2 x^{3}-x^{2}-10 x+5 $$

Step-by-Step Solution

Answer

The polynomial factors to \((x^2 - 5)(2x - 1)\).

Step 1: Identify Pairs

Group the polynomial into two pairs. For the polynomial \(2x^3 - x^2 - 10x + 5\), we group it as: \[ (2x^3 - x^2) + (-10x + 5) \]

Step 2: Factor Each Pair

Factor out the greatest common factor from each pair:For the first pair \((2x^3 - x^2)\), the GCF is \(x^2\), so:\[ x^2(2x - 1) \]For the second pair \((-10x + 5)\), the GCF is \(-5\), so:\[ -5(2x - 1) \]

Step 3: Factor Out the Common Binomial

Notice that both terms now contain a common binomial factor \((2x - 1)\). Factor this out:\[ x^2(2x - 1) - 5(2x - 1) = (x^2 - 5)(2x - 1) \]

Key Concepts

Grouping Method

The grouping method is a powerful technique used for factoring polynomials that have four or more terms. It involves organizing the polynomial into smaller groups, usually pairs, and factoring each pair separately. The aim is to manipulate the expression so that a common factor emerges, which can be further factored out. This method is advantageous because it provides a structured way of reducing complex polynomials into simpler factors.

To apply the grouping method, start by dividing the polynomial into pairs. It's often helpful to look for common variables or coefficients. In our example, the polynomial is broken down into

• \((2x^3 - x^2) + (-10x + 5)\)

This strategic grouping helps in identifying the greatest common factor for each pair, simplifying the solving process.

Common Binomial Factor

After grouping and factoring out the greatest common factors in each pair, the next goal is to identify a common binomial factor. A common binomial factor is an expression that appears in multiple terms, allowing further factorization.

• In our example, after factoring out the greatest common factors, we obtain:
  • \(x^2(2x - 1)\) from the first pair
  • \(-5(2x - 1)\) from the second pair

This common binomial factor, \((2x - 1)\), is crucial because it allows us to consolidate the factored expression into:

• \((x^2 - 5)(2x - 1)\)

Recognizing and factoring out common binomial factors is a key step in simplifying polynomials efficiently.

Greatest Common Factor

The Greatest Common Factor (GCF) is an essential concept when factoring polynomials. It is the largest expression that divides each term of the polynomial without leaving a remainder. Factoring out the GCF simplifies the polynomial and is often the first step in the grouping method.

Consider the polynomial \(2x^3 - x^2 - 10x + 5\). By examining each pair:

• The GCF of \(2x^3\) and \(-x^2\) is \(x^2\), resulting in \(x^2(2x - 1)\) for this pair.
• For \(-10x\) and \(+5\), the GCF is \(-5\), which gives \(-5(2x - 1)\).

Identifying and factoring out the GCF makes it easier to see common factors or binomials that can be extracted further in the process. It's a helpful step towards simplifying complex polynomial expressions.

Factoring Techniques

Factoring techniques are varied methods used to decompose polynomials into products of simpler polynomials. Each technique is useful in different scenarios, and mastering them provides a strong foundation in algebra.

• The grouping method is particularly useful for polynomials with four or more terms, as it involves arranging terms into pairs and factoring them individually.
• Identifying the greatest common factor is the first step in many factoring techniques, simplifying the polynomial before further breakdown.
• The discovery of common binomial factors allows for additional simplification, transforming the polynomial into a product of smaller expressions.

Proficiency with these techniques greatly aids in solving algebraic equations and understanding further mathematical concepts. By practicing different factoring methods, students enhance their problem-solving skills, making them more adept at tackling various mathematical challenges.