Problem 67 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$6 x^{2}+54$$

Step-by-Step Solution

Answer

The polynomial factors to \(6(x^2 + 9)\) and cannot be factored further using integers.

Step 1: Identify a Common Factor

Look for a Greatest Common Factor (GCF) that can be factored out of all terms in the polynomial. Here, notice that both terms, \(6x^2\) and \(54\), are divisible by \(6\).

Step 2: Factor Out the Greatest Common Factor

Factor out the GCF, \(6\), from the polynomial: \(6(x^2 + 9)\).

Step 3: Check for Further Factoring

Examine the expression \(x^2 + 9\) to see if it can be factored further. \(x^2 + 9\) is a sum of squares, which cannot be factored using real integers.

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is a crucial step in simplifying polynomials. The GCF of a set of numbers or terms is the largest number that divides each of the terms evenly. For polynomials, this means finding a common factor for all coefficients. For example, in the given polynomial expression \( 6x^2 + 54 \), we identify the GCF as \( 6 \).

To find this, review each term: \( 6x^2 \) and \( 54 \). Both are divisible by \( 6 \), which means \( 6 \) is the greatest common factor. Factoring the GCF not only simplifies the polynomial but also sets the stage for further factoring processes if applicable.

• Check each term for common numerical factors.
• Use the GCF to rewrite the polynomial in a simpler form.

This simplifies the process and assists in identifying other possible factoring techniques in subsequent steps.

Sum of Squares

The concept of a sum of squares involves polynomial forms like \( a^2 + b^2 \). Unlike a difference of squares, a sum of squares generally does not factor neatly into linear terms using integers. In our polynomial \( x^2 + 9 \), we notice that it represents a sum of squares.

One might look for a shortcut or trick to factor \( x^2 + 9 \), but the reality is that it remains unfactorable over the real numbers using integers. This means:

• Recognizing that some polynomial forms are not factorable in the same ways as others.
• Understanding limitations helps in deciding when a polynomial can be considered completely factored.

This knowledge aids in efficiently navigating through complex polynomial expressions without spending extra time on impossible factorizations.

Factoring Techniques

Factoring techniques involve several strategies to break down polynomials into simpler components. Understanding how to apply different methods correctly helps in tackling a wide range of problems. Here's a quick overview:

• Finding the GCF: Always start by identifying the Greatest Common Factor. It simplifies the polynomial greatly.
• Recognizing patterns: Look for familiar patterns like difference of squares, perfect square trinomials, or other recognizable polynomial forms.
• Sum of squares: As in the case of \( x^2 + 9 \), note that some forms can't be further broken down.

Each technique requires practice to recognize and deploy correctly. By applying these methods, students can systematically and efficiently solve polynomial factoring problems.