Problem 65 - College Algebra

Question

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

Step-by-Step Solution

Answer

The factored form of this polynomial is \(3x(x+1)(x-1)\)

Step 1: Identify the Greatest Common Factor (GCF)

The GCF of \(3x^3\) and \(-3x\) is \(3x\).

Step 2: Factor Out The GCF From Each Term

Factoring out \(3x\) from each term gives \(3x(x^2 - 1)\).

Step 3: Factor the Difference of Squares

The equation \(x^2 - 1\) is a difference of squares equation and can be factored into \((x+1)(x-1)\).

Step 4: Write the Complete Factored Form

We write the complete factored form by multiplying the GCF with the factored difference of squares, getting \(3x(x+1)(x-1)\).

Key Concepts

Greatest Common Factor

Factoring polynomials begins by identifying the Greatest Common Factor (GCF), which is the largest factor shared by each term within the polynomial. When students look at the term 'factoring', they should think of it as a reverse multiplication process—breaking down expressions into simpler parts that can be multiplied to obtain the original expression.

Finding the GCF simplifies the factoring process by reducing the polynomial to an equivalent expression with smaller coefficients and exponents. In our example, the GCF of 3x^3 and -3x is 3x; this is because 3x is the largest expression that divides evenly into both terms.

After we identify the GCF, we 'factor it out' from each term, which essentially involves dividing each term of the polynomial by the GCF and placing the GCF in front of a set of parentheses. The terms inside the parentheses are what's left after the factor of 3x is taken out of each term of the original expression.

Difference of Squares

The Difference of Squares is an essential algebraic concept for students to grasp when factoring certain types of polynomials. It refers to expressions that can be written in the form a^2 - b^2, representing the difference between two perfect squares. Prime examples include x^2 - 1, x^2 - 9, or 4x^2 - 16.

These expressions are special because they can always be factored into the product of two binomials: (a + b)(a - b). For instance, the expression x^2 - 1 can be viewed as the difference between the square of x and the square of 1. By employing the Difference of Squares formula, the factored form of x^2 - 1 becomes (x + 1)(x - 1). This method assists in simplifying equations and solving problems with squared terms.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In our original exercise, 3x^3 - 3x is an algebraic expression with x as the variable. The process of factoring algebraic expressions is much like finding the puzzle pieces that, when combined, will form the given expression.

Algebraic expressions can vary greatly in complexity—ranging from simple monomials like 3x, to binomials such as (x + 1), up to polynomials with many terms. Factoring an algebraic expression requires a fundamental understanding of arithmetic operations, exponent rules, and the application of factoring strategies like finding the GCF and employing the Difference of Squares.

Once the foundational aspects are recognized, algebraic expressions can be factored, simplified, and solved to reveal useful information about the relationships between the variables and constants they contain.