Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 3-27 x^{2} $$

Short answer

3(1 - 3x)(1 + 3x)

Step-by-step solution

- Recognize the form

The given polynomial is in the form 3 - 27x^2. Notice that this is a difference of squares format but with a common factor.

- Factor out the Greatest Common Factor (GCF)

Identify the greatest common factor of the terms 3 and 27x^2. The GCF is 3. Factor out the GCF from the polynomial: 3(1 - 9x^2).

- Simplify inside the parenthesis

Recognize that the expression inside the parenthesis, 1 - 9x^2, is a difference of squares. It can be factored further.

- Apply the difference of squares formula

Use the difference of squares formula, a^2 - b^2 = (a - b)(a + b). Notice that 1 = 1^2 and 9x^2 = (3x)^2, so we can rewrite 1 - 9x^2 as (1 - 3x)(1 + 3x).

- Write the completely factored form

Combine the GCF and the factored form of the expression inside the parenthesis. Therefore, the completely factored form of the polynomial is 3(1 - 3x)(1 + 3x).

Key concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that can evenly divide each term of a polynomial. In our given polynomial, 3 - 27x^2, we first look at the coefficients (3 and 27). Here, 3 is a factor of both 3 and 27.
To factor out the GCF, identify the highest number that divides all coefficients and consider any common variables. Since 27x^2 has no common variables with 3, the GCF is just 3. Factoring out the GCF:

• Divide each term by 3.
• We get 3(1 - 9x^2).

The polynomial is now simplified with its GCF factored out, making it easier to handle in the next steps.

Difference of Squares

The difference of squares is a special form used in polynomial factoring, identified by the format a^2 - b^2.
This format can be rewritten using the formula a^2 - b^2 = (a - b)(a + b). Our expression inside the parenthesis after factoring out the GCF is 1 - 9x^2.
This is indeed a difference of squares where a = 1 and b = 3x. To further factorize:

• 1 is the square of 1, since 1^2 = 1.
• 9x^2 is the square of 3x, since (3x)^2 = 9x^2.

Applying the difference of squares formula: 1 - 9x^2 = (1 - 3x)(1 + 3x).This simplifies our expression inside the parenthesis to two binomials.

Polynomial Simplification

Polynomial Simplification involves breaking down a given polynomial into simpler, more manageable pieces. In our case, starting with 3 - 27x^2, we factored out the Greatest Common Factor (GCF) of 3 first, simplifying it to 3(1 - 9x^2).
Then, using the difference of squares formula, we further simplified 1 - 9x^2 into (1 - 3x)(1 + 3x). Combining these steps:

• First, factor out the GCF: 3(1 - 9x^2).
• Next, apply the difference of squares: 3(1 - 3x)(1 + 3x).

So, the completely factored form of the polynomial is 3(1 - 3x)(1 + 3x). This final form makes it easier to work with in further algebraic operations or solving equations.