To expand $(1 + 2x)^3$ using the binomial theorem, we follow these steps:

The binomial theorem states that for any positive integer $n$:

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

Here, $a = 1$, $b = 2x$, and $n = 3$.

The binomial coefficients $\binom{n}{k}$ are given by:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Now, let's expand $(1 + 2x)^3$:

$(1 + 2x)^3 = \sum_{k=0}^{3} \binom{3}{k} (1)^{3-k} (2x)^k$

We calculate each term of the sum:

• For $k = 0$: $\binom{3}{0} (1)^{3-0} (2x)^0 = \binom{3}{0} \cdot 1^3 \cdot (2x)^0 = 1$

• For $k = 1$: $\binom{3}{1} (1)^{3-1} (2x)^1 = \binom{3}{1} \cdot 1^2 \cdot 2x = 3 \cdot 1 \cdot 2x = 6x$

• For $k = 2$: $\binom{3}{2} (1)^{3-2} (2x)^2 = \binom{3}{2} \cdot 1^1 \cdot (2x)^2 = 3 \cdot 1 \cdot 4x^2 = 12x^2$

• For $k = 3$: $\binom{3}{3} (1)^{3-3} (2x)^3 = \binom{3}{3} \cdot 1^0 \cdot (2x)^3 = 1 \cdot 1 \cdot 8x^3 = 8x^3$

Now, we sum these terms together:

$(1 + 2x)^3 = 1 + 6x + 12x^2 + 8x^3$

Thus, the expanded form of $(1 + 2x)^3$ using the binomial theorem is:

$1 + 6x + 12x^2 + 8x^3$

Would you like more details or have any questions?

Here are 5 questions you might be interested in:

1. What is the binomial theorem and how is it applied in mathematics?
2. How do you calculate binomial coefficients?
3. Can you provide more examples of binomial expansions?
4. How is the binomial theorem used in probability and statistics?
5. What are some applications of the binomial theorem in real life?

Tip: Understanding the binomial theorem is helpful not only for expanding binomials but also for solving problems in combinatorics, probability, and algebra.