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Difficulty: EasyFactoring Polynomials

Which of the following represents the completely factored form of the algebraic expression 3x2−273x^2 - 27?

  1. 3(x−3)(x+3)3(x - 3)(x + 3)Answer
  2. B
    3(x2−9)3(x^2 - 9)
  3. C
    3(x−3)(x−3)3(x - 3)(x - 3)
  4. D
    3x(x−9)3x(x - 9)

Answer

The completely factored form is 3(x−3)(x+3)3(x - 3)(x + 3).
The expression is completely factored by first extracting the greatest common factor of 3 to get 3(x2−9)3(x^2 - 9), and then factoring the resulting difference of squares into (x−3)(x+3)(x - 3)(x + 3).

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from all terms in the expression.
3(x2−9)3(x^2 - 9)
Both terms share a common factor of 3. Factoring this out simplifies the remaining polynomial.
2
Examine the expression inside the parentheses to see if it can be factored further.
x2−9x^2 - 9 is a difference of squares.
The expression x2−9x^2 - 9 follows the pattern a2−b2a^2 - b^2, where a=xa = x and b=3b = 3.
3
Factor the difference of squares using the standard algebraic pattern (a−b)(a+b)(a - b)(a + b).
(x−3)(x+3)(x - 3)(x + 3)
Expanding (x−3)(x+3)(x - 3)(x + 3) yields x2+3x−3x−9x^2 + 3x - 3x - 9, which simplifies back to x2−9x^2 - 9.
4
Combine the GCF and the factored binomials to write the final completely factored form.
3(x−3)(x+3)3(x - 3)(x + 3)
An expression is only completely factored when all of its factors are prime (cannot be factored further).

Key Concept

Factoring by finding a Greatest Common Factor (GCF) followed by factoring a difference of squares.
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Topics

Factoring Polynomials

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