find the coefficient of x³ in the expansion of (2x² + x - 3)⁴

To find the coefficient of \(x^3\) in the expansion of \((2x^2 + x - 3)^4\), we will use the **binomial theorem** and **multinomial expansion**.

We are expanding:
\[
(2x^2 + x - 3)^4
\]

We need to consider the terms that contribute to \(x^3\). In a multinomial expansion, each term in the product comes from selecting one element from each factor of \((2x^2 + x - 3)\), which is done four times.

### Possible Combinations of Terms:

Each term contributing to \(x^3\) will come from a combination of choosing powers of \(2x^2\), \(x\), and \(-3\) from the four factors, and the sum of their exponents of \(x\) must equal 3.

We have three relevant cases:
1. Select \(2x^2\) once, \(x\) once, and \(-3\) twice.
2. Select \(2x^2\) zero times, \(x\) three times, and \(-3\) once.

Now let's calculate each contribution.

#### Case 1: Select \(2x^2\) once, \(x\) once, and \(-3\) twice.
The term will be:
\[
(2x^2)^1 \cdot (x)^1 \cdot (-3)^2 = 2x^2 \cdot x \cdot 9 = 18x^3
\]
The number of ways to choose which of the four terms comes from \(2x^2\), \(x\), and \(-3\) is calculated by the multinomial coefficient:
\[
\binom{4}{1, 1, 2} = \frac{4!}{1!1!2!} = \frac{24}{2} = 12
\]
So, the contribution from this case is:
\[
12 \times 18x^3 = 216x^3
\]

#### Case 2: Select \(x\) three times and \(-3\) once.
The term will be:
\[
(x)^3 \cdot (-3)^1 = x^3 \cdot (-3) = -3x^3
\]
The number of ways to choose which of the four terms comes from \(x\) and \(-3\) is:
\[
\binom{4}{3, 1} = \frac{4!}{3!1!} = 4
\]
So, the contribution from this case is:
\[
4 \times (-3x^3) = -12x^3
\]

### Total Contribution:
Adding up both cases, the total coefficient of \(x^3\) is:
\[
216 + (-12) = 204
\]

Thus, the coefficient of \(x^3\) in the expansion of \((2x^2 + x - 3)^4\) is **204**.

Would you like more details on any part of the process or any additional questions?

Here are some related questions you might find interesting:
1. How does the multinomial theorem work for higher powers of binomials?
2. What would the expansion look like for \((x^2 + 2x - 1)^5\)?
3. How can we generalize this process for any trinomial raised to a power?
4. How does the binomial expansion compare to the multinomial expansion?
5. Can you find the constant term in the expansion of \((2x^2 + x - 3)^4\)?

**Tip**: When expanding polynomials with more than two terms, always focus on selecting combinations that match the required degree (such as \(x^3\) in this case) to simplify the process.

Find the coefficient of x³ in the expansion of (2x² + x - 3)⁴

This solution shows how to find the coefficient of x³ in the expansion of (2x² + x - 3)⁴ using the multinomial theorem and polynomial expansion techniques. Aimed at students in Grades 11-12, the solution includes detailed steps and the application of key algebraic concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation