Answer :

To factor the polynomial [tex]\(3x^3 - 5x^2 - 27x + 45\)[/tex] completely, we'll identify and extract its factors.

Here's a step-by-step guide to understanding the factorization:

1. Identify a Potential Factor:
We begin by looking for potential rational roots using the Rational Root Theorem. However, in this case, we'll provide the factorization directly.

2. Check for Simplified Factors:
The polynomial factors are [tex]\((x - 3)\)[/tex], [tex]\((x + 3)\)[/tex], and [tex]\((3x - 5)\)[/tex]. Let's verify these factors make up the original polynomial by multiplication.

3. Verify by Multiplication:
Multiply the factors to check if it reconstructs the original polynomial.

First, multiply [tex]\((x - 3)\)[/tex] and [tex]\((x + 3)\)[/tex]:
[tex]\[
(x - 3)(x + 3) = x^2 + 3x - 3x - 9 = x^2 - 9
\][/tex]

Next, multiply the result with [tex]\((3x - 5)\)[/tex]:
[tex]\[
(x^2 - 9)(3x - 5) = x^2 \cdot 3x + x^2 \cdot (-5) - 9 \cdot 3x - 9 \cdot (-5)
\][/tex]
Simplify each term:
[tex]\[
= 3x^3 - 5x^2 - 27x + 45
\][/tex]

This verifies our factorization.

4. Final Factorization:
The polynomial [tex]\(3x^3 - 5x^2 - 27x + 45\)[/tex] factors completely as:
[tex]\[
(x - 3)(x + 3)(3x - 5)
\][/tex]

This completes the factorization process, confirming that we have factored the given polynomial correctly.

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