The coefficients in a binomial expansion are determined using the binomial theorem, which states:
$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
Here, $\binom{n}{k}$ is the binomial coefficient, calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Let’s expand $(x - 2)^5$ step by step.

---

Step 1: Recognize the Binomial Form

$(x - 2)^5$ can be written as:
$(x + (-2))^5$
Here, $a = x$, $b = -2$, and $n = 5$.

---

Step 2: Use the Binomial Theorem Formula

The expansion is:
$(x - 2)^5 = \sum_{k=0}^5 \binom{5}{k} x^{5-k}(-2)^k$

This means you calculate each term by substituting $k = 0, 1, 2, \dots, 5$.

---

Step 3: Calculate the Binomial Coefficients

For each $k$, the binomial coefficient $\binom{5}{k}$ is:
$\binom{5}{k} = \frac{5!}{k!(5-k)!}$
Where $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$. Let’s calculate for each $k$:

• $k = 0: \binom{5}{0} = \frac{5!}{0!5!} = 1$
• $k = 1: \binom{5}{1} = \frac{5!}{1!4!} = 5$
• $k = 2: \binom{5}{2} = \frac{5!}{2!3!} = 10$
• $k = 3: \binom{5}{3} = \frac{5!}{3!2!} = 10$
• $k = 4: \binom{5}{4} = \frac{5!}{4!1!} = 5$
• $k = 5: \binom{5}{5} = \frac{5!}{5!0!} = 1$

So, the coefficients are: $1, 5, 10, 10, 5, 1$.

---

Step 4: Apply the Formula for Each Term

The general term is:
$\binom{5}{k} x^{5-k} (-2)^k$
Let’s compute each term:

1. $k = 0: \binom{5}{0} x^{5-0}(-2)^0 = 1 \cdot x^5 \cdot 1 = x^5$
2. $k = 1: \binom{5}{1} x^{5-1}(-2)^1 = 5 \cdot x^4 \cdot (-2) = -10x^4$
3. $k = 2: \binom{5}{2} x^{5-2}(-2)^2 = 10 \cdot x^3 \cdot 4 = 40x^3$
4. $k = 3: \binom{5}{3} x^{5-3}(-2)^3 = 10 \cdot x^2 \cdot (-8) = -80x^2$
5. $k = 4: \binom{5}{4} x^{5-4}(-2)^4 = 5 \cdot x^1 \cdot 16 = 80x$
6. $k = 5: \binom{5}{5} x^{5-5}(-2)^5 = 1 \cdot x^0 \cdot (-32) = -32$

---

Step 5: Write the Final Expansion

Combine all terms:
$(x - 2)^5 = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32$

---

Key Takeaway:

The coefficients come from the binomial coefficients $\binom{5}{k}$, which are determined by factorials. The powers of $x$ and $-2$ depend on the index $k$.

---

Would you like additional examples or clarification?

---

5 Related Questions:

1. What happens when you expand $(x + 3)^4$?
2. How do binomial coefficients relate to Pascal's Triangle?
3. Why does the power of $x$ decrease while the power of $(-2)$ increases?
4. Can you generalize the expansion for any $(x + a)^n$?
5. How do negative exponents affect the expansion?

Tip: