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Chapter 0: Problem 65

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

Short Answer

Expert verified
The completely factored form of the polynomial \(3x^{3} - 3x\) is \(3x(x - 1)(x + 1)\).

Step by step solution

01

Find the greatest common factor (GCF)

The first step in factorizing any expression is to find the greatest common factor (GCF) of all the terms. For the given polynomial \(3x^{3} - 3x\), the GCF is \(3x\).
02

Apply the distributive property

In the second step, apply the distributive property to factor out the GCF. It involves dividing each term of the polynomial by the GCF. After dividing, we get \(x^{2} - 1\). Therefore, \(3x^{3} - 3x = 3x(x^{2} - 1)\).
03

Factoring the difference of squares

In the final step, identify \(x^{2} - 1\) as a difference of squares which can also be factorized. Factoring \(x^{2} - 1\) gives \((x - 1)(x + 1)\). Hence, the completely factored form of \(3x^{3} - 3x\) is \(3x(x - 1)(x + 1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor
The Greatest Common Factor (GCF) is a fundamental concept in factoring polynomials. It is the largest factor that divides all terms of the polynomial without leaving a remainder. Finding the GCF is the first step in simplifying or factoring polynomials.To find the GCF of a polynomial like \(3x^{3} - 3x\), follow these steps:
  • Identify the common numerical factor. Here, both terms of the polynomial, \(3x^{3}\) and \(3x\), share the number 3.
  • Identify the common variable factor. Both terms have the variable \(x\), with the smallest power being \(x^{1}\).
  • Multiply the common numerical and variable factors. For this example, the GCF is \(3x\).
Recognizing and factoring out the GCF not only simplifies the problem but also prepares the polynomial for further factorization. It's a crucial technique that makes tackling complex polynomials easier, much like breaking down large tasks into manageable parts.
Distributive Property
The distributive property allows us to factor out the GCF we previously identified. It involves rewriting an expression as a product by distributing the GCF across each term of the polynomial.For instance, the polynomial \(3x^{3} - 3x\) can be written in terms of its GCF:
  • Divide each term of the expression by the GCF, \(3x\), which simplifies \(3x^{3}\) to \(x^{2}\) and \(-3x\) to \(-1\).
  • Express the polynomial as a product: \(3x(x^{2} - 1)\).
Using the distributive property in factoring helps in simplifying expressions by isolating common factors. It's like grouping similar items together, making the expression neater and easier to manipulate further.
Difference of Squares
The concept of the difference of squares is a straightforward yet powerful tool in polynomial factoring. It applies to expressions of the form \(a^{2} - b^{2}\), which factors into \((a - b)(a + b)\).For the expression \(x^{2} - 1\), recognize it as a difference of squares:
  • Here, \(a^{2}\) is \(x^{2}\) and \(b^{2}\) is 1, as 1 is the square of itself.
  • This expression factors into \((x - 1)(x + 1)\).
The simplicity of this method lies in its ability to quickly dismantle expressions into their prime components. Whether it's reviewing for exams or solving challenging algebraic problems, recognizing and factoring differences of squares can be a significant advantage.

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