find the coeficient of x6 in (3x2-5)power 7

We are asked to find the coefficient of \( x^6 \) in the expansion of \( (3x^2 - 5)^7 \). To solve this, we can use the **binomial theorem**, which states:

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

In our case:
- \( a = 3x^2 \)
- \( b = -5 \)
- \( n = 7 \)

We want to find the term that contains \( x^6 \). In each term of the expansion, the power of \( x \) is determined by \( (3x^2)^{n-k} \). Specifically, we need to find the term where the power of \( x \) is 6.

1. The general term in the expansion is:
\[
T_k = \binom{7}{k} (3x^2)^{7-k} (-5)^k
\]
Simplifying \( (3x^2)^{7-k} \):
\[
T_k = \binom{7}{k} 3^{7-k} x^{2(7-k)} (-5)^k
\]

We need \( x^{2(7-k)} = x^6 \), which gives:
\[
2(7-k) = 6 \quad \Rightarrow \quad 7-k = 3 \quad \Rightarrow \quad k = 4
\]

So, we substitute \( k = 4 \) into the general term:
\[
T_4 = \binom{7}{4} 3^{3} x^6 (-5)^4
\]

Now calculate:
\[
T_4 = \binom{7}{4} \cdot 27 \cdot x^6 \cdot 625
\]
\[
T_4 = 35 \cdot 27 \cdot 625 \cdot x^6
\]
\[
T_4 = 35 \cdot 16875 \cdot x^6
\]
\[
T_4 = 590625 \cdot x^6
\]

Thus, the coefficient of \( x^6 \) is **590625**.

---

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the binomial theorem and when is it used?
2. How can we find the coefficient of other powers of \( x \) in binomial expansions?
3. Can you explain the meaning of \( \binom{n}{k} \) in the binomial expansion?
4. What if we were asked to find the coefficient of \( x^4 \) instead?
5. How does the negative sign in \( (-5) \) affect the binomial expansion?

**Tip:** When expanding powers using the binomial theorem, always keep track of both the power of \( x \) and the numerical coefficients carefully to avoid errors.

find the coefficient of x^6 in (3x^2 - 5)^7

This detailed solution walks you through finding the coefficient of x^6 in the binomial expansion of (3x^2 - 5)^7. The problem uses the binomial theorem and algebraic principles to calculate the coefficient. Suitable for 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