Problem 68 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 3 x^{3}-27 x $$

Step-by-Step Solution

Answer

\(3x(x - 3)(x + 3)\)

Step 1: Identify the Common Factor

Look at the terms in the expression \(3x^3 - 27x\). Notice that both terms have a common factor of \(3x\). This is the greatest common factor that we will factor out of the expression.

Step 2: Factor out the Common Factor

Factor \(3x\) out of the expression. Divide each term by \(3x\):\[3x^3 \div 3x = x^2\]\[-27x \div 3x = -9\]This gives us:\[3x(x^2 - 9)\]

Step 3: Recognize the Difference of Squares

The expression inside the parentheses \(x^2 - 9\) is a difference of squares. Recall the identity for the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\). In this case, \(x^2 - 9\) can be rewritten as \((x - 3)(x + 3)\) because \(9 = 3^2\).

Step 4: Factor the Difference of Squares

Apply the difference of squares formula to factor \(x^2 - 9\):\[(x^2 - 9) = (x - 3)(x + 3)\]Substitute back into the expression:\[3x(x - 3)(x + 3)\]

Step 5: Verify Your Solution

To ensure correctness, distribute and multiply the factors back out. \((x - 3)(x + 3)\) gives \(x^2 - 9\), and multiplying by \(3x\) gives the original expression \(3x^3 - 27x\). Therefore, the completely factored expression is:\[3x(x - 3)(x + 3)\]

Key Concepts

Greatest Common Factor

When factoring polynomials, the greatest common factor (GCF) is the largest factor that all terms in the polynomial share. Finding the GCF is typically the first step in the factoring process because it simplifies the polynomial, making it easier to work with.
To find the GCF, examine each term of the polynomial to see what numerical and variable factors they have in common. In the exercise given, the polynomial is \(3x^3 - 27x\). Both terms, \(3x^3\) and \(-27x\), can be divided by \(3x\), which becomes our GCF.
Here are the steps to identify and factor out the GCF:

• Look at the coefficients (3 and -27) and take the largest number that divides both without leaving a remainder, which is 3.
• Look at the variable part. All terms have \(x\), so take the smallest degree of \(x\), which is \(x^1\).
• Combine these to form \(3x\) as the GCF, and divide each term by \(3x\) to factor it out.

Once the GCF is factored out, the polynomial becomes simple enough for further factoring if needed.

Difference of Squares

The difference of squares is a specific algebraic pattern that is common in polynomial expressions. It appears as \(a^2 - b^2\) which can be decomposed into \((a-b)(a+b)\). Understanding this concept is essential for recognizing patterns that can be factored easily.
In our polynomial, after factoring out the GCF, we are left with \(x^2 - 9\). Notice that \(x^2\) is a perfect square and 9 is also a perfect square \((3^2)\). Thus, \(x^2 - 9\) fits the pattern of a difference of squares:\[ a^2 - b^2 = (a-b)(a+b) \] where \(a = x\) and \(b = 3\).
This pattern helps us to quickly factor out \(x^2 - 9\) into \((x - 3)(x + 3)\). These two binomial factors are fully factored parts of the original expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and operation symbols. They do not have an equality sign, unlike equations. Factoring these expressions helps simplify them, making it easier to perform operations or solve when they're part of a larger algebraic equation.
Every algebraic expression we work with can often be decomposed into simpler factors, as illustrated in the main exercise. This particular expression, \(3x^3 - 27x\), initially appears complex. However, by applying the principles of GCF and the difference of squares, we simplify it into a product of three factors: \(3x(x - 3)(x + 3)\).
Factoring involves recognizing patterns and mathematical identities (such as the difference of squares), that allow you to manipulate and rearrange terms into friendlier forms. This process is foundational to algebraic problem solving, as it assists in reduction and analysis of expressions within equations or more complex mathematical contexts.