Problem 76 - College Algebra Essentials

Question

Factor completely, or state that the polynomial is prime. $$9 x^{3}-9 x$$

Step-by-Step Solution

Answer

The fully factored form of the given polynomial is \(9x \cdot (x-1) \cdot (x+1)\)

Step 1: Identify Common Factor

The first step in factoring an expression is to look for a common factor in all terms. Here, both terms \(9 x^{3}\) and \(-9 x\) have a common factor of \(9x\). Let's factor out \(9x\)

Step 2: Factor Out The Greatest Common Factor

By factoring out \(9x\), the polynomial now becomes \(9x \cdot (x^{2}-1)\). We now have the original polynomial expressed as a product, but the expression within the brackets can still be factored.

Step 3: Apply Difference of Squares Formula

In the expression \(x^{2}-1\), we have the difference of squares which can be factored using the formula \(a^{2}-b^{2}=(a+b)(a-b)\). Here, \(a=x\) and \(b=1\). Hence, we can write \(x^{2}-1\) as \((x-1)(x+1)\)

Step 4: Write The Final Factored Form

By applying the previous steps, the final factored form of the given polynomial is \(9x \cdot (x-1) \cdot (x+1)\)

Key Concepts

Greatest Common Factor

To understand polynomial factoring, we start with the greatest common factor, also known as the GCF. In algebra, the GCF is the largest expression that divides all the terms in the polynomial without leaving a remainder.

Identifying the GCF is crucial because it simplifies the polynomial and makes further factoring more manageable. For example, in the polynomial \(9x^3 - 9x\), both terms share a common factor of \(9x\). Removing the GCF, \(9x\), results in a new expression \(9x(x^2 - 1)\).

To find the GCF, follow these steps:

• List the factors of each term in the polynomial.
• Determine the highest factor common to all terms.

These steps will help efficiently break down any expression, setting the stage for smoother factoring of more complicated polynomials.

Difference of Squares

The difference of squares is a specific pattern in polynomial expressions that allows for straightforward factoring. It is characterized by expressions of the form \(a^2 - b^2\). This pattern simplifies into two binomials: \((a + b)(a - b)\).

In the step by step example, the expression inside the brackets \(x^2 - 1\) fits this pattern precisely. Here, \(a\) is \(x\), and \(b\) is \(1\), transforming \(x^2 - 1\) into \((x + 1)(x - 1)\). Recognizing this pattern is important because it drastically simplifies the polynomial.

Utilizing the difference of squares:

• Identify if your expression matches \(a^2 - b^2\).
• Write it as \((a + b)(a - b)\).

This method not only simplifies but also ensures no factors are overlooked, maximizing efficiency and correctness in polynomial factoring.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like + and -). These expressions form the core of algebra and, when manipulated through factoring, reveal simpler or more useful forms.

Factoring is a key tool for simplifying algebraic expressions, as shown in the given example. By factoring \(9x^3 - 9x\), we not only found a form easier to work with but also uncovered the underlying expressions: \((x - 1)(x + 1)\). This new form is beneficial for solving equations, simplifying fractions, and more.

When working with algebraic expressions:

• Look for opportunities to factor out the greatest common factor.
• Be on the lookout for recognizable patterns like the difference of squares.

These steps facilitate the transformation of complex polynomials into simpler expressions, enhancing both understanding and problem-solving capabilities in algebra.