https://file.aiquickdraw.com/mathbot/file/1727035482839im9l1730.jpg

The problem asks for the constant term in the binomial expansion of the expression:

\[
\left( 2x^2 - \frac{1}{2x} \right)^{30}
\]

### Solution Approach

To find the constant term, we need to apply the binomial theorem, which is used to expand expressions of the form \((a + b)^n\). The general term in the expansion is given by:

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

Here, \(a = 2x^2\), \(b = -\frac{1}{2x}\), and \(n = 30\).

#### General Term

The \(k\)-th term in the expansion is:

\[
T_k = \binom{30}{k} \left(2x^2\right)^{30-k} \left(-\frac{1}{2x}\right)^k
\]

Simplifying each factor:
- \(\left( 2x^2 \right)^{30-k} = 2^{30-k}x^{2(30-k)} = 2^{30-k}x^{60-2k}\)
- \(\left(-\frac{1}{2x}\right)^k = (-1)^k \frac{1}{2^k x^k} = (-1)^k \frac{1}{2^k} x^{-k}\)

So the general term becomes:

\[
T_k = \binom{30}{k} (-1)^k \frac{2^{30-k}}{2^k} x^{60-2k-k} = \binom{30}{k} (-1)^k 2^{30-2k} x^{60-3k}
\]

### Finding the Constant Term

For the term to be a constant, the exponent of \(x\) must be zero. That is:

\[
60 - 3k = 0 \quad \Rightarrow \quad k = 20
\]

### Substituting \(k = 20\)

Substitute \(k = 20\) into the general term:

\[
T_{20} = \binom{30}{20} (-1)^{20} 2^{30-2(20)} x^{60-3(20)}
\]
\[
T_{20} = \binom{30}{20} (1) 2^{30-40} = \binom{30}{20} 2^{-10}
\]

Thus, the constant term is:

\[
\boxed{\binom{30}{20} \cdot \frac{1}{2^{10}}}
\]

You are not required to compute the binomial coefficient itself, so this is the final expression for the constant term.

Would you like further details or have any questions?

---

Here are five related questions:
1. How do we apply the binomial theorem to higher powers of expressions?
2. Why does the exponent of \(x\) need to be zero for a term to be constant?
3. Can you explain the pattern of signs when expanding binomials with negative terms?
4. How do we calculate binomial coefficients, and why are they important in expansions?
5. What are some other techniques for finding specific terms in expansions?

**Tip**: Always simplify powers and coefficients step-by-step in binomial expansions to avoid errors in complex calculations.

Beräkna den konstanta termen i utvecklingen av (2x^2 - 1/(2x))^30. Svaret kan innehålla en binomialkoefficient som du inte behöver räkna ut värdet på.

Learn how to find the constant term in the binomial expansion of (2x^2 - 1/(2x))^30 using the binomial theorem. This algebraic problem explores binomial coefficients and powers of x, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation