Factor the polynomial expression 8x3+18x3+1.

A. (2x+1)(4x2−2x+1)(2x+1)(4x2−2x+1)

B. (2x−1)(4x2+x+1)(2x−1)(4x2+x+1)

C. (2x+1)(4x2+x−1)(2x+1)(4x2+x−1)

D. (2x+1)(4x2+x+1)


The given expression is \( 8x^3 + 18x^2 + 1 \).

To factor this polynomial, let's analyze it step by step.

### Step 1: Look for common factors
There are no common factors among the terms, so we proceed to factoring by grouping or trial methods.

### Step 2: Check for possible factorization patterns
Since the expression is cubic, we consider the possibility of factoring it into a product of binomials and trinomials.

### Step 3: Trial and error for binomial and trinomial factors
Let's try to factor it as \((2x + a)(4x^2 + bx + c)\). 

Expanding \((2x + a)(4x^2 + bx + c)\), we get:

\[
2x(4x^2 + bx + c) + a(4x^2 + bx + c) = 8x^3 + 2bx^2 + 2cx + 4ax^2 + abx + ac
\]

Comparing this to \(8x^3 + 18x^2 + 1\), we need to find values of \(a\), \(b\), and \(c\) such that:

\[
8x^3 + (2b + 4a)x^2 + (2c + ab)x + ac = 8x^3 + 18x^2 + 1
\]

By comparing coefficients:

- \(2b + 4a = 18\)
- \(ac = 1\)
- \(2c + ab = 0\)

### Step 4: Test the given options
Now, let's test the provided options by expanding them and comparing them to the original expression.

- **Option A:** \((2x + 1)(4x^2 - 2x + 1)\)

    Expanding this:

    \[
    (2x + 1)(4x^2 - 2x + 1) = 8x^3 - 4x^2 + 2x + 4x^2 - 2x + 1 = 8x^3 + 1
    \]
    This doesn't match the original expression.

- **Option B:** \((2x - 1)(4x^2 + x + 1)\)

    Expanding this:

    \[
    (2x - 1)(4x^2 + x + 1) = 8x^3 + 2x^2 + 2x - 4x^2 - x - 1 = 8x^3 - 2x^2 + x - 1
    \]
    This doesn't match the original expression.

- **Option C:** \((2x + 1)(4x^2 + x - 1)\)

    Expanding this:

    \[
    (2x + 1)(4x^2 + x - 1) = 8x^3 + 2x^2 - 2x + 4x^2 + x - 1 = 8x^3 + 6x^2 - x - 1
    \]
    This doesn't match the original expression.

- **Option D:** \((2x + 1)(4x^2 + x + 1)\)

    Expanding this:

    \[
    (2x + 1)(4x^2 + x + 1) = 8x^3 + 2x^2 + 2x + 4x^2 + x + 1 = 8x^3 + 6x^2 + 3x + 1
    \]

None of these options seem to match exactly upon testing. There might be a typo or misunderstanding in the provided options. The correct factorization needs to be found by trying different combinations or revisiting the expression.

Please confirm if there was a typo or would you like further assistance with a different approach?

Here are some related questions to consider:

1. How do you factor polynomials by grouping?
2. What are the special factorization formulas?
3. How can we verify the factorization of a polynomial?
4. What techniques can be used to factor cubic polynomials?
5. What is the difference between factorization and expanding?

**Tip:** When factoring polynomials, always double-check by expanding to ensure your factors are correct.

Learn how to factor the polynomial expression 8x^3 + 18x^2 + 1 with a detailed step-by-step solution. Explore methods for factoring cubic polynomials and check the correctness of factorization. Suitable for high school students and those preparing for advanced algebra.

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