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Chapter 5: Problem 72

Factor each polynomial completely. $$3 x^{3}-27 x$$

Short Answer

Expert verified
The completely factored form is \(3x(x - 3)(x + 3)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Find the greatest common factor of the terms in the polynomial. The terms are \(3x^3\) and \(-27x\). The GCF is \(3x\).
02

Factor Out the GCF

Factor out the greatest common factor from the polynomial. This gives us: \(3x(x^2 - 9)\).
03

Recognize the Difference of Squares

Notice that the expression \(x^2 - 9\) is a difference of squares. Recall that \(a^2 - b^2 = (a - b)(a + b)\). Here \(a = x\) and \(b = 3\).
04

Factor the Difference of Squares

Apply the difference of squares formula. \(3x(x^2 - 9) = 3x(x - 3)(x + 3)\).
05

Write the Final Factored Form

The completely factored form of the polynomial is \(3x(x - 3)(x + 3)\).

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Key Concepts

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

Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is a critical concept in polynomial factorization. It is the largest factor shared by all terms in the polynomial.
To find the GCF, follow these steps:
  • List the factors of each term.
  • Select the largest common factor.
For example, let鈥檚 consider the polynomial from the exercise: \(3x^3 - 27x\). First, identify the factors of each term:
  • For \(3x^3\), the factors are 3, x, x, and x.
  • For \(-27x\), the factors are -27, x, and its numerical factorization (-27 = -3 脳 3 脳 3).
The GCF is the largest common factor present in both terms. In this case, the factor both 3 and x appear in both terms, giving us a GCF of \(3x\). Factoring out the GCF, we get: \[3x(x^2 - 9)\].
Difference of Squares
The Difference of Squares is a specific factoring pattern in which a polynomial can be expressed as two squared terms subtracted from one another, i.e., \(a^2 - b^2\).
When identified, it can be factored into: \[(a-b)(a+b)\].
Let鈥檚 apply this to our exercise's polynomial:
After factoring out the GCF, we have \(3x(x^2 - 9)\). Notice that \(x^2 - 9\) is a difference of squares:
  • \(x^2 = (x)^2\)
  • \(9 = (3)^2\)
So, using the difference of squares formula, we rewrite \(x^2 - 9\) as: \[(x - 3)(x + 3)\]. Adding back the GCF from our previous step, the polynomial is now factored as: \[3x(x - 3)(x + 3)\].
Algebraic Expressions
Understanding Algebraic Expressions is essential for mastering polynomial factorization. An algebraic expression is a combination of variables, numbers, and operators. Consider the parts in our polynomial:
  • Coefficians are 3 and -27
  • Variables are x
  • Exponentials like \(x^3\)
Factorizing involves breaking down complex expressions into simpler, multiplicative factors. The process typically involves:
  • Finding the GCF
  • Recognizing special patterns (like difference of squares)
  • Rewriting the expression in simpler terms
For the exercise鈥檚 polynomial \(3x^3 - 27x\), the steps were:
  1. Identifying the GCF which is \(3x\)
  2. Factoring out the GCF: \[3x(x^2 - 9)\]
  3. Finding the difference of squares: \[x^2 - 9 = (x - 3)(x + 3)\]
  4. Combining all factors: \[3x(x - 3)(x + 3)\]
This approach helps simplify algebraic expressions into their most basic form, facilitating a deeper understanding and easier manipulation.

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