Answer :

We start with the polynomial

[tex]$$
27x^3 + 9x^2 - 75x - 25.
$$[/tex]

Step 1. Factor by grouping:

Group the terms into two pairs:

[tex]$$
(27x^3 + 9x^2) + (-75x - 25).
$$[/tex]

In the first group, factor out the common factor [tex]$9x^2$[/tex]:

[tex]$$
27x^3 + 9x^2 = 9x^2(3x + 1).
$$[/tex]

In the second group, factor out [tex]$-25$[/tex]:

[tex]$$
-75x - 25 = -25(3x + 1).
$$[/tex]

Now, the polynomial becomes:

[tex]$$
9x^2(3x + 1) - 25(3x + 1).
$$[/tex]

Since both terms contain the common factor [tex]$(3x + 1)$[/tex], factor it out:

[tex]$$
(3x + 1)(9x^2 - 25).
$$[/tex]

Step 2. Factor the quadratic as a difference of squares:

Notice that

[tex]$$
9x^2 - 25 = (3x)^2 - 5^2,
$$[/tex]

which is a difference of squares. Recall that

[tex]$$
a^2 - b^2 = (a + b)(a - b).
$$[/tex]

So, we factor it as:

[tex]$$
9x^2 - 25 = (3x + 5)(3x - 5).
$$[/tex]

Step 3. Write the complete factorization:

Substitute the factorization of [tex]$9x^2 - 25$[/tex] back into the expression:

[tex]$$
(3x + 1)(9x^2 - 25) = (3x + 1)(3x + 5)(3x - 5).
$$[/tex]

Thus, the completely factored form of the polynomial is:

[tex]$$
\boxed{(3x + 1)(3x + 5)(3x - 5)}.
$$[/tex]

Other Questions