Test questions for finding specific terms and coefficients in binomial expansions

📚 Quick Study Guide

• 🔢 The Binomial Theorem: $(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k$, where ${n \choose k} = \frac{n!}{k!(n-k)!}$
• 🔍 General Term: The $(r+1)^{th}$ term in the expansion of $(a + b)^n$ is given by ${n \choose r} a^{n-r} b^r$.
• 💡 Finding Specific Terms: To find a specific term (e.g., the term independent of $x$), set up the general term and solve for $r$.
• 📝 Finding Coefficients: The coefficient of a term is the numerical factor multiplying the variables. It is given by ${n \choose r}$.
• ➕ Pascal's Triangle: Provides a quick way to find binomial coefficients for small values of $n$. Each number is the sum of the two numbers above it.
• 🧪 Properties: The sum of the exponents of $a$ and $b$ in each term is always equal to $n$. The coefficients are symmetrical.

Practice Quiz

1. What is the coefficient of $x^3$ in the expansion of $(x + 2)^5$?
1. A) 10
2. B) 40
3. C) 80
4. D) 32
2. Find the term independent of $x$ in the expansion of $(2x + \frac{1}{x})^6$.
1. A) 15
2. B) 60
3. C) 160
4. D) 240
3. What is the sum of the coefficients in the expansion of $(3x - 2y)^4$?
1. A) 1
2. B) 16
3. C) 81
4. D) 625
4. Determine the middle term in the expansion of $(x^2 - 1)^{10}$.
1. A) $252x^{10}$
2. B) $-252x^{10}$
3. C) $210x^{10}$
4. D) $-210x^{10}$
5. Find the coefficient of $x^2$ in the expansion of $(1 - x)^8$.
1. A) 7
2. B) 28
3. C) -28
4. D) 56
6. What is the constant term in the expansion of $(x + \frac{2}{x^2})^3$?
1. A) 0
2. B) 6
3. C) 8
4. D) 12
7. Determine the coefficient of $x$ in the expansion of $(1 + x + x^2)^5$.
1. A) 5
2. B) 10
3. C) 15
4. D) 20