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To determine whether \(2x - 3\) is a factor of \(8x^3 + 16x^2 - 18x - 36\), we can use polynomial division or the Remainder Theorem.

### Remainder Theorem Approach:
If we substitute the zero of \(2x - 3 = 0\), which is \(x = \frac{3}{2}\), into the polynomial and the result is zero, then \(2x - 3\) is a factor.

Let’s substitute \(x = \frac{3}{2}\) into the polynomial \(8x^3 + 16x^2 - 18x - 36\):

\[
P(x) = 8x^3 + 16x^2 - 18x - 36
\]

1. \(8\left(\frac{3}{2}\right)^3 = 8 \times \frac{27}{8} = 27\)
2. \(16\left(\frac{3}{2}\right)^2 = 16 \times \frac{9}{4} = 36\)
3. \(-18 \times \frac{3}{2} = -27\)
4. \(-36\)

Now, summing these values:
\[
P\left(\frac{3}{2}\right) = 27 + 36 - 27 - 36 = 0
\]

Since \(P\left(\frac{3}{2}\right) = 0\), the remainder is 0, and \(2x - 3\) **is** a factor of \(8x^3 + 16x^2 - 18x - 36\).

Would you like a more detailed explanation of the steps involved?

### Here are some related questions for further exploration:
1. What is synthetic division, and how can it be applied to this problem?
2. How does polynomial division confirm whether a given polynomial is a factor of another?
3. What is the significance of the Remainder Theorem in factorization?
4. How do you find the roots of a cubic polynomial?
5. Can all cubic polynomials be factored over the real numbers?

**Tip**: When using the Remainder Theorem, always substitute the value of \(x\) that makes the divisor zero to check if the remainder is 0.

Is 2x - 3 a factor of 8x^3 + 16x^2 - 18x - 36?

Determine whether 2x - 3 is a factor of the polynomial 8x^3 + 16x^2 - 18x - 36 using the Remainder Theorem. This detailed step-by-step solution walks through the substitution of x = 3/2 to check if the remainder is zero.

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