The expression you provided is:

$2x^3 - 5x^2 - 5x - 6$

Step 1: Check for common factors

First, we check if there's a common factor across all terms. In this case, there isn't any common factor, so we move on to factorization.

Step 2: Factor by grouping

Since this is a cubic polynomial, one method to factor it is by grouping:

1. Group the terms in pairs:

   $(2x^3 - 5x^2) - (5x + 6)$

2. Factor out the common factors from each group:

   From the first group $2x^3 - 5x^2$, factor out $x^2$: $x^2(2x - 5)$

   From the second group $-5x - 6$, factor out $-1$ to make the signs inside the parentheses match: $-1(5x + 6)$

   So the expression becomes: $x^2(2x - 5) - (5x + 6)$

Step 3: Combine and check for further factorization

At this point, the expression doesn't seem to factor nicely by grouping. So, let's check if we can factor the cubic polynomial by trying to find a rational root using the Rational Root Theorem.

Step 4: Apply the Rational Root Theorem

The Rational Root Theorem suggests that any rational root $p/q$ of the polynomial, where $p$ is a factor of the constant term $-6$ and $q$ is a factor of the leading coefficient $2$, can be checked. Possible rational roots are:

$\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}$

We can test these values by substituting them into the polynomial to see if they yield zero.

For example:

• Try $x = 2$:

$2(2)^3 - 5(2)^2 - 5(2) - 6 = 16 - 20 - 10 - 6 = -20 \quad \text{(Not a root)}$

• Try $x = -1$:

$2(-1)^3 - 5(-1)^2 - 5(-1) - 6 = -2 - 5 + 5 - 6 = -8 \quad \text{(Not a root)}$

After testing the possible roots, let's try $x = -2$ since the positive ones didn't work.

• Try $x = -2$:

$2(-2)^3 - 5(-2)^2 - 5(-2) - 6 = -16 - 20 + 10 - 6 = -32 + 10 - 6 = 0 \quad \text{(Root found!)}$

Step 5: Factor out $x + 2$

Since $x = -2$ is a root, $x + 2$ is a factor. We can divide the polynomial by $x + 2$ using synthetic or long division.

Performing synthetic division:

• The quotient after dividing $2x^3 - 5x^2 - 5x - 6$ by $x + 2$ is:

$2x^2 - 9x + 3$

So, the complete factorization is:

$(2x^3 - 5x^2 - 5x - 6) = (x + 2)(2x^2 - 9x + 3)$

Final Answer:

The factorized form of the expression is:

$(2x^3 - 5x^2 - 5x - 6) = (x + 2)(2x^2 - 9x + 3)$

Would you like more details or have any questions?

Here are some related questions:

1. How does the Rational Root Theorem help in factoring polynomials?
2. What is synthetic division, and how is it used?
3. How do you factor quadratic polynomials after finding a root?
4. What if no rational roots are found?
5. Can you use grouping for all cubic polynomials?
6. How do you identify the degree of a polynomial?
7. How can factoring be used to solve polynomial equations?
8. What are some tips for factoring higher-degree polynomials?

Tip: When factoring cubic polynomials, always check for possible rational roots first; it simplifies the process significantly.