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The problem asks to find the standard form of the polynomial \( p(x) = (2x - 3)^4 \). Let's expand the binomial expression step by step to determine which option represents the expanded form correctly.

### Step 1: Use the Binomial Theorem
The binomial expansion of \( (a - b)^n \) is given by:

\[
(a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k
\]

In this case, \( a = 2x \), \( b = 3 \), and \( n = 4 \). Using the binomial theorem:

\[
(2x - 3)^4 = \sum_{k=0}^{4} \binom{4}{k} (2x)^{4-k} (-3)^k
\]

### Step 2: Expand the Terms
We will calculate the terms of the expansion:

1. \( k=0 \): \( \binom{4}{0} (2x)^4 (-3)^0 = 1 \cdot 16x^4 = 16x^4 \)
2. \( k=1 \): \( \binom{4}{1} (2x)^3 (-3)^1 = 4 \cdot 8x^3 \cdot (-3) = -96x^3 \)
3. \( k=2 \): \( \binom{4}{2} (2x)^2 (-3)^2 = 6 \cdot 4x^2 \cdot 9 = 216x^2 \)
4. \( k=3 \): \( \binom{4}{3} (2x)^1 (-3)^3 = 4 \cdot 2x \cdot (-27) = -216x \)
5. \( k=4 \): \( \binom{4}{4} (2x)^0 (-3)^4 = 1 \cdot 1 \cdot 81 = 81 \)

### Step 3: Combine the Terms
Now, combine all the terms to get the expanded polynomial:

\[
p(x) = 16x^4 - 96x^3 + 216x^2 - 216x + 81
\]

### Step 4: Match the Options
The expanded polynomial is:

\[
p(x) = 16x^4 - 96x^3 + 216x^2 - 216x + 81
\]

This matches the last option in the list.

Thus, the correct answer is the last option:

\[
\boxed{p(x) = 16x^4 - 96x^3 + 216x^2 - 216x + 81}
\]

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Would you like further details or explanations on this? Here are some related questions for practice:

1. How does the binomial theorem work for a general binomial expansion?
2. What is the role of binomial coefficients in expansions like this?
3. Can you expand \( (x + 2)^3 \) using the same method?
4. How would the result change if \( (2x - 3)^3 \) was expanded instead of \( (2x - 3)^4 \)?
5. What are some practical applications of polynomial expansions in fields like physics or economics?

**Tip:** For problems involving polynomial expansion, the binomial theorem is a quick and efficient method!

The polynomial p is defined as p(x) = (2x - 3)^4. Which of the following is p(x) written in standard form?

Find the standard form of the polynomial expression p(x) = (2x - 3)^4 using the binomial theorem. This step-by-step guide covers binomial expansions, key formulas, and algebraic techniques. Suitable for students in Grades 9-12.

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