Problem 57 - College Algebra

Question

In Exercises \(57-84\), factor completely, or state that the polynomial is prime. $$3 x^{3}-3 x$$

Step-by-Step Solution

Answer

After factoring completely, the polynomial \(3x^{3}-3x\) becomes \(3x(x-1)(x+1)\).

Step 1: Identify Common Factor

Observe the polynomial \(3x^{3}-3x\). Both terms are divisible by a common factor. In this case, the common factor is \(3x\).

Step 2: Factor Out The Common Factor

Next, we separate the common factor. Doing so, we get \(3x(x^{2}-1)\).

Step 3: Factor The Difference of Squares

Here, \(x^{2}-1\) is a difference of squares which is a special case in factoring. Any expression in the form \(a^{2} - b^{2}\) can be factored as \((a + b)(a - b)\). Applying that rule, we factor \(x^{2}-1\) as \((x-1)(x+1)\).

Step 4: Write The Completely Factored Form

Replace \(x^{2}-1\) with \((x-1)(x+1)\) in the expression \(3x(x^{2}-1)\). We get the completely factored form as \(3x(x-1)(x+1)\).

Key Concepts

Common Factor in Polynomial Factoring

When we are faced with a polynomial such as 3x^3 - 3x, the process of simplifying it by factoring often begins by identifying a common factor. This is a number or expression that evenly divides each term of the polynomial. In our example, each term can be divided by 3x, making it the common factor.
Finding the common factor simplifies the problem significantly. After we divide each term of the polynomial by 3x, we're left with x^2 - 1. This extraction process is a crucial step and aids in revealing more intricate factorization opportunities that lie within the remaining polynomial expression.
To visualize this, think of the polynomial as a piece of clay. Finding and removing the common factor is like stripping away an outer layer, uncovering the detailed shapes hidden underneath that may lead us to further simplification through other factoring techniques.

Difference of Squares

Within the realm of factoring, the difference of squares represents a particularly elegant case. It refers to an expression that can be written in the form a^2 - b^2, which can then be factored into the product of two binomials: (a + b)(a - b).

Detecting a Difference of Squares

In the factored polynomial 3x(x^2 - 1), we can spot x^2 - 1 as a classic example of a difference of squares. Breaking it down, x^2 is our a^2, and 1 is our b^2. We apply our rule and express x^2 - 1 as (x - 1)(x + 1).
It’s the subtraction sign that’s crucial here; a sum of squares, in contrast, doesn’t factor similarly. Students should be vigilant for this pattern, as recognizing a difference of squares can unlock a shortcut to factoring complex expressions.

Factoring by Grouping

Sometimes, a polynomial doesn't neatly present a common factor or a difference of squares. In those cases, factoring by grouping might be the tool you need. While this technique wasn’t required for the given exercise since we had only two terms, it's worthwhile to understand it for more complex polynomials.
Factoring by grouping involves arranging a polynomial into groups—usually pairs—of terms that have a common factor. You then factor out the common factor from each group, which may reveal a common binomial factor. Finally, factoring out that binomial gives you the fully factored form of the original polynomial.
For polynomials with four or more terms, this method can be particularly effective. You would group the terms into twos, factor each pair individually, and then look for a common factor. Keep in mind that sometimes rearranging the terms before grouping can reveal the best way to factor the polynomial.