Question

If factored completely, \(3 x^{3}-12 x=\) __________.

Short answer

3x(x + 2)(x - 2)

Step-by-step solution

Identify the Greatest Common Factor (GCF)

First, observe the given expression: \[3x^3 - 12x\] Identify the greatest common factor (GCF) of the terms in the expression. Both terms, \(3x^3\) and \(-12x\), share a common factor of \(3x\).

Factor out the GCF

Factor out the GCF \(3x\) from each term in the expression: \[3x(x^2 - 4)\]

Identify the Difference of Squares

Now, observe the expression inside the parenthesis: \[x^2 - 4\] This is a difference of squares, which can be factored further into \((x + 2)(x - 2)\).

Complete the Factorization

Combine the factored form of the difference of squares with the GCF factored out earlier: \[3x(x + 2)(x - 2)\]

Key concepts

greatest common factor

When factoring polynomials, one of the first steps is identifying the **Greatest Common Factor (GCF)**. The GCF is the largest factor that divides all terms in the polynomial without leaving a remainder. It simplifies the polynomial and makes further factorization easier.

In the expression \[3x^3 - 12x\], identify the GCF of \[3x^3\] and \[-12x\].
The factors of \[3x^3\] are \[3, x, x^2, x^3\].
The factors of \[-12x\] are \[-12, -6, -4, -3, -2, -1, x\].
By comparison, you find that \[3x\] is the greatest factor common to both terms.
Applying the GCF allows us to simplify the expression:
\[3x(x^2 - 4)\]

This simplification makes the next steps in factoring much easier.

difference of squares

After using the GCF, you often find expressions like \[x^2 - 4\]. Recognizing patterns like these as a **Difference of Squares** is crucial. The difference of squares is a special polynomial property where an expression takes the form \[a^2 - b^2\].

This can be factored into \[(a + b)(a - b)\].
In our example, \[x^2 - 4\], we see:
\[a = x\] and \[b = 2\]
Thus, it simplifies to:
\[(x + 2)(x - 2)\]

This pattern shows up frequently, so recognizing it can save time and simplify your work.

polynomial factorization

Factoring polynomials involves breaking them down into simpler 'factors'. These make solving equations and simplifying expressions easier. There's a basic approach for polynomial factorization:

**Step-by-step guide to polynomial factorization:**

• Identify the GCF: Find the largest common factor of all the terms.
• Factor out the GCF: Rewrite the polynomial by factoring out the GCF.
• Simplify remaining expression: Look for patterns or recognize special polynomials like the difference of squares.
• Factor further if possible: Continue factoring until you can't factor anymore.

For the given exercise, by combining these steps:
\[3x^3-12x\] leading to \[3x(x^2-4)\]
then recognizing \[x^2-4\] as a difference of squares to factor into \[(x+2)(x-2)\]
gives the final factorization of \[3x(x+2)(x-2)\].

Mastering these techniques makes polynomial problems more approachable and manageable.