Answer :

To completely factor the polynomial [tex]\( 27x^3 + 9x^2 - 75x - 25 \)[/tex], we can follow a structured approach to simplify and break down the expression into its factors. Here's a step-by-step guide on how to achieve that:

1. Identify Groups for Factoring:
- The given polynomial is [tex]\( 27x^3 + 9x^2 - 75x - 25 \)[/tex]. Notice that there are two sets of terms here that can be grouped: [tex]\( (27x^3 + 9x^2) \)[/tex] and [tex]\( (-75x - 25) \)[/tex].

2. Factor by Grouping:
- First, factor each group separately.

First Group: [tex]\( 27x^3 + 9x^2 \)[/tex]
- In the first group, factor out the common factor, which is [tex]\( 9x^2 \)[/tex]:
[tex]\[
9x^2(3x + 1)
\][/tex]

Second Group: [tex]\( -75x - 25 \)[/tex]
- In the second group, factor out the common factor, which is [tex]\(-25\)[/tex]:
[tex]\[
-25(3x + 1)
\][/tex]

3. Combine the Groups:
- After factoring both groups, you have:
[tex]\[
9x^2(3x + 1) - 25(3x + 1)
\][/tex]
- Notice that [tex]\( (3x + 1) \)[/tex] is a common factor in both groups.

4. Factor out the Common Binomial Factor:
- Factor out the common binomial [tex]\((3x + 1)\)[/tex]:
[tex]\[
(3x + 1)(9x^2 - 25)
\][/tex]

5. Recognize and Factor the Difference of Squares:
- The expression [tex]\( 9x^2 - 25 \)[/tex] is a difference of squares:
[tex]\[
9x^2 - 25 = (3x)^2 - (5)^2
\][/tex]
- It can be factored further as:
[tex]\[
(3x - 5)(3x + 5)
\][/tex]

6. Combine All Factors:
- Finally, combine all factors:
[tex]\[
(3x + 1)(3x - 5)(3x + 5)
\][/tex]

So, the completely factored form of the polynomial [tex]\( 27x^3 + 9x^2 - 75x - 25 \)[/tex] is:
[tex]\[
(3x - 5)(3x + 1)(3x + 5)
\][/tex]

This factorization shows how the original cubic polynomial can be broken down into the product of three linear factors.

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