Problem 103 - Elementary Algebra

Question

Factor out the GCF first and then factor by grouping. $$ 2 x 3+6 x 2-10 x-30 $$

Step-by-Step Solution

Answer

The factored form is \(2(x + 3)(x^2 - 5)\).

Step 1: Identify the GCF

Examine each term in the polynomial \(2x^3, 6x^2, -10x, -30\) to find the greatest common factor (GCF). The common factor for all terms is 2.

Step 2: Factor Out the GCF

Factor out the GCF, which is 2, from each term of the polynomial: \(2(x^3 + 3x^2 - 5x - 15)\).

Step 3: Group the Polynomial

Look at the polynomial \(x^3 + 3x^2 - 5x - 15\). Group the terms to make factorings easier: \((x^3 + 3x^2) + (-5x - 15)\).

Step 4: Factor by Grouping

Factor each group separately. From \((x^3 + 3x^2)\), factor out \(x^2\) to get \(x^2(x + 3)\). From \((-5x - 15)\), factor out \(-5\) to get \(-5(x + 3)\).

Step 5: Combine Factored Groups

Recognize that \(x+3\) is a common factor: \(x^2(x+3) - 5(x+3)\). This allows factoring by grouping the terms: \((x + 3)(x^2 - 5)\).

Step 6: Combine All Factors

Combine everything: the factored GCF and the grouped factors. The fully factored form is \(2(x + 3)(x^2 - 5)\).

Key Concepts

Greatest Common Factor (GCF)

Understanding the Greatest Common Factor (GCF) is a key step in simplifying and factoring polynomials. The GCF of a set of terms is the largest factor that all the terms share. Finding this factor simplifies the expression significantly before additional factoring methods are applied.

To find the GCF, examine each term's numerical coefficient and variable parts:

• For coefficients, find the largest number that divides all coefficients evenly.
• For variables, use the lowest power of the variable that is common to all terms.

In our example, the polynomial is composed of the terms: \(2x^3, 6x^2, -10x, -30\).
The numerical coefficients (2, 6, -10, and -30) share a common factor of 2.
Hence, 2 is the GCF, as it is the largest whole number that can evenly divide each of these coefficients.

Finding the GCF and factoring it out reduces the polynomial, making it easier to handle in subsequent steps.

Factor by Grouping

After identifying and factoring out the GCF, the next step in simplifying complex polynomials is known as "Factor by Grouping." This method involves splitting the polynomial into groups that can be separately factored, followed by combining these factors.

Here are the steps for efficient factor by grouping:

• Split the polynomial into two groups. Aim for groups that share a common factor.
• In the given polynomial \(x^3 + 3x^2 - 5x - 15\), group the terms as \((x^3 + 3x^2) + (-5x - 15)\).
• Factor out the common factor in each group.

From \((x^3 + 3x^2)\), \(x^2\) is factored out, resulting in \(x^2(x + 3)\).
For \((-5x - 15)\), factor out \(-5\), giving \(-5(x + 3)\).
Now, notice that \(x+3\) is a common factor between the groups, which allows further simplification.

Recognizing a common factor in both groups and factoring it out is the essence of factoring by grouping. This yields a final simpler factored expression.

Polynomial Expressions

Polynomial expressions comprise a sum of terms, each consisting of a constant coefficient and a variable raised to a power. The structure of a polynomial allows for a variety of operations, including addition, subtraction, multiplication, and factoring, which are vital in both algebraic simplifications and real-world applications.

Key Characteristics of Polynomial Expressions:

• The terms are limited to non-negative integer powers (e.g., \(2x^3, 6x^2\)).
• They can have one or multiple variables, although basic operations typically involve one main variable.
• Polynomial expressions are arranged in standard form, often in descending powers.

When looking at a polynomial, understanding its structure helps you determine the best approach to solving or factoring it.
In the case of our polynomial \(2x^3 + 6x^2 - 10x - 30\), starting with the GCF and proceeding to factor by grouping simplifies the expression effectively.

Developing a solid grasp of polynomial fundamentals aids in managing complex expressions and applying advanced algebraic techniques with ease.