2x^{3}-16x^{2}-18xcompletely using the XBOX method.


The XBOX method (or Box method) is a factoring technique often used for trinomials or polynomials of higher degrees. Here’s how we factor the cubic polynomial \( 2x^3 - 16x^2 - 18x \) completely using this method:

### Step-by-step process:

#### 1. Factor out the greatest common factor (GCF):
The first step is to factor out the GCF from the polynomial. The GCF of \( 2x^3 \), \( -16x^2 \), and \( -18x \) is \( 2x \).

So, we begin by factoring out \( 2x \):

\[
2x(x^2 - 8x - 9)
\]

Now, we need to factor \( x^2 - 8x - 9 \).

#### 2. Use the XBOX method to factor the quadratic \( x^2 - 8x - 9 \):

The XBOX method starts with identifying the coefficients:
- The coefficient of \( x^2 \) (the leading coefficient) is \( 1 \).
- The middle term coefficient is \( -8 \).
- The constant term is \( -9 \).

We need to find two numbers that multiply to give the product of \( 1 \times -9 = -9 \) and add to give \( -8 \). 

These two numbers are \( -9 \) and \( 1 \), since:
- \( -9 \times 1 = -9 \)
- \( -9 + 1 = -8 \)

#### 3. Rewrite the middle term:
Using these two numbers, we rewrite the middle term \( -8x \) as \( -9x + 1x \):

\[
x^2 - 9x + x - 9
\]

#### 4. Group the terms:
Now, we group the terms in pairs:

\[
(x^2 - 9x) + (x - 9)
\]

#### 5. Factor each group:
Factor out the common factor from each group:

\[
x(x - 9) + 1(x - 9)
\]

#### 6. Factor the common binomial:
Both terms have a common binomial factor of \( (x - 9) \). So, factor that out:

\[
(x + 1)(x - 9)
\]

#### 7. Combine with the GCF:
Now, don’t forget the \( 2x \) we factored out at the start. Thus, the fully factored form of \( 2x^3 - 16x^2 - 18x \) is:

\[
2x(x + 1)(x - 9)
\]

### Final Answer:
\[
2x(x + 1)(x - 9)
\]

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Would you like more details on any of these steps? Here are some questions you could consider:

1. How do I find the greatest common factor (GCF) of more complicated polynomials?
2. What other methods can be used to factor quadratic expressions?
3. How does the XBOX method differ from other factoring methods like the AC method?
4. Can the XBOX method be used for higher-degree polynomials as well?
5. Why do we group terms in the middle during the XBOX method?

**Tip**: Always check your factorization by multiplying the factors back together to make sure they result in the original polynomial.

2x^{3}-16x^{2}-18x completely using the XBOX method.

This detailed solution explains how to factor the polynomial 2x^3 - 16x^2 - 18x using the XBOX method. The process involves factoring out the greatest common factor (GCF) and applying the XBOX method to factor the quadratic expression. Suitable for students in Grades 9-12.

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